The Cheeger problem in abstract measure spaces

We consider non-negative $\sigma$-finite measure spaces coupled with a proper functional $P$ that plays the role of a perimeter. We introduce the Cheeger problem in this framework and extend many classical results on the Cheeger constant and on Cheeger sets to this setting, requiring minimal assumptions on the pair measure space-perimeter. Throughout the paper, the measure space will never be asked to be metric, at most topological, and this requires the introduction of a suitable notion of Sobolev spaces, induced by the coarea formula with the given perimeter.


Introduction
In the Euclidean framework, the Cheeger constant of a given set Ω ⊂ R n is defined as where L n (E) and P (E) respectively denote the n-dimensional Lebesgue measure of E and the variational perimeter of E. In the past decades, the Cheeger constant has been extensively studied and generalized in several research directions in view of its many applications: the lower bounds to the first eigenvalue of the Dirichlet p-Laplacian operator [82], and the equivalence of such an inequality with Poincaré's [102] (up to some convexity assumptions); the relation with the torsion problem [32,33]; the existence of sets with prescribed mean curvature [5,90]; the existence of graphs with prescribed mean curvature [73,89]; the reconstruction of noisy images [58,109]; the minimum flowmaximum cut problem [75,113], with medical applications [16].In addition, the Cheeger constant has been employed in elastic-plastic models of plate failure [85], and some of its weighted variants have found applications to Bingham fluids and landslide models [79].Quite surprisingly, the Cheeger constant of a square even appears in a theoretic elementary proof of the Prime Number Theorem [17]!Because of its numerous applications, several authors have been drawn to the subject and started to investigate the same problem and these links also in frameworks different from the usual Euclidean one.Actually, the constant was first introduced in a Riemannian setting as a way to bound from below the first eigenvalue of the Laplace-Beltrami operator [59].The argument proposed by Cheeger is sound and robust, as noticed even earlier by Maz'ya [98,99] (an English translation is available in [76]) and is apt to be ported to several other settings.
In the settings mentioned above, the proofs mostly follow those available in the usual Euclidean space.In the present paper, we are interested in pinpointing the minimal assumptions needed on the space and on the perimeter functional in order to establish the fundamental properties of Cheeger sets, as well as the links to other problems.In the following, we shall be interested in non-negative σ-finite measure spaces endowed with a perimeter functional satisfying some suitable assumptions.
Our approach fits into a broader current of research that has gained popularity in the past decade, aiming to study variational problems, well-known in the Euclidean setting, in general spaces under the weakest possible assumptions.Quite often, the ambient space is a (metric) measure spaces.For example, such a general point of view has been adopted for the variational mean curvature of a set [20], for shape optimization problems [37], for Anzellotti-Gauss-Green formulas [74], for the total variation flow [34,35] and, very recently, for the existence of isoperimetric clusters [107].
According to our best knowledge, we tried to give the most comprehensive bibliography, in the spirit of [110].Any suggestion on missing relevant papers is welcomed.
1.1.Structure of the paper and results.In Section 2, we introduce the basic setting of perimeter σ-finite measure spaces, that is, non-negative σ-finite measure spaces (X, A , m) endowed with a proper functional P : A → [0, +∞] possibly satisfying suitable properties (P.1)-(P.7)that we shall require from time to time.
A considerable effort goes towards defining BV functions in measure spaces, where a metric is not available.Indeed, usually, the perimeter functional is induced by the metric.In our setting, instead, only a perimeter functional is at disposal, so we use it to define BV functions by defining the total variation via the coarea formula with the given perimeter.
To properly define Sobolev functions, we need a slightly richer structure, requiring the measure space to be endowed with a topology, and the perimeter functional P (•) to arise from a relative (w.r.t.open sets A) perimeter functional P ( • ; A).By using the relative perimeter, we refine the notion of BV function by requiring that the variation is a finite measure.This, in turn, allows us to define W 1,1 functions as BV functions with variation being absolutely continuous with respect to the reference measure.Afterwards, via relaxation, we can define W 1,p functions for any p ∈ (1, +∞).For a more detailed overview, we refer the reader to Section 1.1.3and Section 1. 1.4.Once the general framework is set, we then shall start to tackle the problem of our interest.
1.1.1.Definition and existence.In Section 3, we define the Cheeger constant of a set Ω in terms of the ratio of the perimeter functional and the measure the space is endowed with.Actually, in a more general vein similar to that of [47], we shall define the N-Cheeger constant as where, as usual, a N-cluster is a N-tuple of pairwise disjoint subsets of Ω, each of which with positive finite measure and finite perimeter.In Theorem 3.1 we provide a general existence result.Unsurprisingly, the key assumptions on the perimeter are the lower-semicontinuity and the compactness of its sublevels with respect to the L 1 norm, besides a isoperimetric-type property that prevents minimizing sequences to converge towards sets with null m-measure.
Further, we provide inequalities between the N-Cheeger and M-Cheeger constants and prove some basic properties of N-Cheeger sets, with a particular attention to the case N = 1.
1.1.2.Link to sets with prescribed mean curvature.In Section 4, we introduce the notion of P -mean curvature as an extension of the usual variational mean curvature to measure spaces that do not possess a metric structure, in the spirit of [20].With this notion at our disposal, we show that any 1-Cheeger set has h 1 (Ω) as one of its P -mean curvatures, see Corollary 4.5.An analogous result holds for the chambers of a N-cluster minimizing h N (Ω), see Corollary 4. 6.
In Theorem 4.7, we investigate the link between h 1 (Ω) and the existence of non-trivial minimizers of the prescribed P -curvature functional where κ is a fixed positive constant, among subsets of Ω.Such a functional, again requiring the lower-semicontinuity and the L 1 -compactness of sublevel sets of the perimeter, has minimizers.If, additionally, one assumes that the perimeter functional satisfies P (∅) = 0, then h 1 (Ω) acts as a threshold for the existence of non-trivial minimizers, that is, for κ < h 1 (Ω) negligible sets are the only minimizers, while for κ > h 1 (Ω) non-trivial minimizers exist.1.1.3.Link to the first eigenvalue of the Dirichlet 1-Laplacian.In the Euclidean space, one defines the first eigenvalue of the Dirichlet 1-Laplacian in a variational way as the infimum . (1.1) In Section 5, we investigate the relation between the 1-Cheeger constant and a suitable reformulation of the constant λ 1,1 (Ω) in our abstract context.In the Euclidean setting [82] and, actually, in the more general anisotropic central-symmetric Euclidean setting [83], the constant λ 1,1 (Ω) coincides with h 1 (Ω) provided that the boundary of the set Ω is sufficiently smooth (e.g., Lipschitz regular).In particular, one can equivalently consider either smooth functions or BV -regular functions.Moreover, because of the smoothness of the boundary of Ω, it holds that BV (Ω) = BV 0 (Ω), where BV 0 (Ω) = {u ∈ BV (R n ) : u = 0 a.e. on R n \ Ω}.Thus, under some regularity assumptions on Ω, one can equivalently restate the problem in (1.1) as In a general set Ω in the Euclidean space, the infimum in (1.2) is less than or equal to that in (1.1), since one only has the inclusion BV (Ω) ⊂ BV 0 (Ω).
In a (possibly non-metric) perimeter-measure space, the constant λ 1,1 (Ω) has to be suitably defined, since neither a notion of derivative (needed to state (1.1)) nor integration-by-parts formulas (needed to define BV functions and thus state (1.2)) are at disposal.To overcome this difficulty, we adopt the usual point of view [57,118,119] and define the total variation of a function via the (generalized) coarea formula provided that the function t → P ({u > t}) is L 1 -measurable, and define the relevant BV space as that of those L 1 functions with finite total variation.For more details, we refer the reader to our Section 2.2.This approach allows us to consider problem (1.2) without any underlying metric structure.In addition, no regularity of the set Ω is required, since there is no need for the problem (1.2) to be equivalent to its regular counterpart (1.1) which, in the present abstract framework, cannot be even formally stated.
With this notion of total variation at hand, we prove that the constant λ 1,1 (Ω) coincides with the 1-Cheeger constant h 1 (Ω) under minimal assumptions on the perimeter, that is, we require that the perimeter of negligible sets and of the whole space is zero, the perimeter is lower-semicontinuous with respect to the L 1 norm, and that the perimeter of a set coincides with that of its complement set, see Theorem 5.4.Moreover, we prove some inequalities relating the N-Cheeger constant h N (Ω) with a cluster counterpart of (1.2).
As observed in Remark 5.9, if one slightly modifies the definition of λ 1,1 (Ω) by considering non-negative functions as the only competitors, then one can obtain the relation with the Cheeger constant even for perimeter functionals which are not symmetric with respect to the complement-set operation.
1.1.4.Link to the p-Dirichlet-Laplacian and the p-torsion.In the Euclidean space, the 1-Cheeger constant comes into play in estimating some quantities related to Laplace equation and to the torsional creep equation.More precisely, it provides lower bounds to the first eigenvalue of the Dirichlet p-Laplacian for p > 1 and to the L 1 norm of the p-torsional creep function.In Section 6 we extend these results to our more general framework.
Both these problems require an extensive preliminary work to define Sobolev spaces in our general (non-metric) context.In order to do so we need a little more structure on the perimeter-measure space: we require it to be endowed with a topology, we require the class of measurable sets to be that of Borel sets, and we require the perimeter P (•) to stem from a relative perimeter when evaluated relatively to the whole space X.
We here quickly sketch how we construct these Sobolev spaces, and we refer the interested reader to our Section 2.3.A relative perimeter functional allows, again via the relative coarea formula in a similar fashion to (1.3), to define the relative variation of a L 1 function u with respect to a measurable set.When this happens to define a measure, we shall say that the function is BV(X, m), and this extends the notion briefly discussed in Section 1.1.3and formally introduced in Section 2.2.When this measure happens to be absolutely continuous with respect to m, we shall say that the function is W 1,1 (X, m) and that the density of the measure with respect to m is the 1slope of u.Via approximation arguments, one can define then the p-slope of a function and the associated W 1,p (X, m) spaces.In turn, the approximation properties allow to define the Sobolev space W 1,p 0 (Ω, m), refer to Definition 6.1.Summing up, Sobolev spaces can be built as induced by a relative perimeter on the topological measure space.Once this notion is available one can define the first eigenvalue of the Dirichlet p-Laplacian for p > 1 in an analogous manner to the standard, Euclidean one.In the classical setting, similarly to (1.1), one defines In our setting we cannot directly consider (1.4) since no notion of derivative is available.Notice, though, that the natural space of competitors of such a problem, is the classic space of W 1,p 0 (Ω) functions, and we do have an analogous notion of Sobolev space at our disposal and thus such a way is viable.
In Euclidean settings [82,83], it is known that the inequality holds.In Theorem 6.3 and Corollary 6.4 we prove that this inequality naturally extends to our general framework, provided that the relative perimeter satisfies some general assumptions.Finally, we recall that the p-torsional creep function is the solution of the PDE with homogeneous Dirichlet boundary datum where −∆ p is the p-Laplace operator.It is known [32] that the solution w p of the PDE (1.6) satisfies As usual, we cannot directly consider (1.6), but we can work with the underlying Euler-Lagrange energy among functions in the Sobolev spaces we defined.In particular, we can prove that minimizers of the energy, if they exist, satisfy (1.7) up to a slightly worse prefactor of p 1+ 1 p , refer to Theorem 6.5, provided that the relative perimeter satisfies some very general properties.
1.1.5.Examples.In the last section of the paper, we collect several examples of spaces that meet our hypotheses.In particular, our very general approach basically covers all results known so far about the existence of Cheeger sets in finite dimensional spaces, and the relation of the constant with the first eigenvalue of the Dirichlet p-Laplacian in numerous contexts.In some of the frameworks presented in Section 7, the results are new, up to our knowledge.
Unfortunately, our approach does not cover the case of the infinite-dimensional Wiener space.In this case, one can suitably define the Cheeger constant and prove the existence of Cheeger sets.Nonetheless, this requires ad hoc notions of BV function and of perimeter which are quite different from the ones adopted in the present paper.We refer the interested reader to [50,Sect. 6] for a more detailed exposition about this specific framework.The present research has been started during a visit of the first and fourth author at the Mathematics Department of the University of Trento.The authors have worked on the present paper during the XXXI Convegno Nazionale di Calcolo delle Variazioni held in Levico Terme (Italy), supported by the the Mathematics Department of the University of Trento and the Centro Internazionale per la Ricerca Matematica (CIRM).The authors wish to thank these institutions for their support and kind hospitality.

Perimeter-measure spaces
The basic setting is that of non-negative σ-finite measure spaces (X, A , m).We set that, for any A, B ∈ A , by A ⊂ B we mean that m(A \ B) = 0. We also let L 0 (X, m) be the vector space of m-measurable functions and, for p ≥ 1, we let L p (X, m) be the usual space of p-integrable functions, that is, As usual, in case X is endowed with a topology T ⊂ P(X), we let B(X) be the Borel σ-algebra generated by T and, in this case, we shall assume that A = B(X).
We remark that all the properties listed above will appear every now and then throughout the paper, but they are not enforced throughout -every statement will precisely contain the bare minimum for its validity.Remark 2.1 (P is invariant under m-negligible modifications).Let property (P.4) be in force.If A, B ∈ A are such that m(A△B) = 0, then P (A) = P (B).To see this, consider any measurable set E and any m-negligible set N, look at the constant sequence {E ∪N} k converging to E in L 1 (X, m), and the constant one {E} k converging to E ∪ N and exploit (P.4).Remark 2.2.Let property (P.6) be in force.If P (E) = 0, then the set E is mnegligible, that is, m(E) = 0. Conversely, if m(E) > 0, then P (E) ∈ (0, +∞].Thus, property (P.6) says that the only sets with finite measure that could possibly have zero perimeter are m-negligible sets.Moreover, if property (P.1) and (P.4) are in force as well, then m-negligible sets have zero perimeter, thanks to Remark 2.1.

N-Cheeger constant and N-Cheeger sets.
We now introduce the central notions of the present paper.
Remark 2.7.By definition, as Ω is required to be N-admissible, the N-Cheeger constant of Ω is finite.Moreover, by Remark 2.5, so it is h M (Ω) for all integers M such that M ≤ N. We refer also to Proposition 3.5.
2.2.Variation and BV functions.We define the variation of a function u ∈ L 0 (X, m) as otherwise. ( With this notation at hand, we let be the set of L 1 functions with bounded variation. We begin with the following result, proving that assuming the validity of properties (P.1) and (P.2), the variation coincides with the perimeter functional on characteristic functions.Lemma 2.8 (Total variation of sets).Let properties (P.1) and (P.2) be in force.If E ∈ A , then Var(χ E ) = P (E).
Proof.The proofs of the first three points are natural consequences of the definition.
Proof of (iv).Let u k , u ∈ L 1 (X, m) be such that u k → u in L 1 (X, m) as k → +∞.Without loss of generality, we can assume that lim inf k→+∞ Var(u k ) < +∞, so that, up to possibly passing to a subsequence (which we do not relabel for simplicity), we have Var(u k ) < +∞ for all k ∈ N. Following [93,Rem. 13.11], one has thus, we immediately deduce that χ {u k >t} → χ {u>t} in L 1 (X, m) as k → +∞ for L 1 -a.e.t ∈ R. Thanks to property (P.4), we have that for L 1 -a.e.t ∈ R, and the map t → P ({u > t}) is also L 1 -measurable.Therefore by Fatou's Lemma we conclude that proving (iv).
The following result, which can be proved as in [57] up to minor modifications, states that the variation functional is convex as soon as the perimeter functional is sufficiently well-behaved.Proposition 2.11 (Convexity of variation).Let properties (P.1), (P.2), (P.3) and (P.4) be in force.Then, Var :

Relative perimeter and relative variation.
In this subsection, we assume that the set X is endowed with a topology T such that A = B(X), the Borel σ-algebra generated by T .
In the Borel measure space (X, B(X), m), we consider a functional and we call it the relative perimeter.In the following, the relative perimeter given in (2.5) may satisfy some assumptions that we list below: (RP.1)P(∅; A) = 0 for all A ∈ T ; (RP.2) P(X; A) = 0 for all A ∈ T ; (RP.3) P(E ∩ F ; A) + P(E ∪ F ; A) ≤ P(E; A) + P(F ; A) for all E, F ∈ B(X) and A ∈ T ; (RP.4) for each A ∈ T , P(• ; A) is lower-semicontinuous with respect to the (strong) convergence in L 1 (X, m).We stress that in the properties above, the perimeter is relative to an open set A, and not to a general element of the Borel σ-algebra.
Moreover, following the same idea of Section 2.2, we let be the variation of u ∈ L 0 (X, m) relative to A ∈ B(X).In analogy with the approach developed in the previous sections, for each A ∈ T one can regard the map P(• ; A) : B(X) → [0, +∞] as a particular instance of the perimeter functional introduced in (2.1).Specifically, we use the notation for all E ∈ B(X) and u ∈ L 0 (X, m) and we consider P (E) as the perimeter of E in the sense of Section 2.1.Analogously, Var(u) stands as the variation of u in the sense of Section 2.2.Consequently, the space Adopting the usual notation, if E ∈ B(X) has finite perimeter measure, then we write P(E; A) = |Dχ E |(A) for all A ∈ B(X).Similarly, if u ∈ L 0 (X, m) has finite variation measure, then we write Var(u; A) = |Du|(A) for all A ∈ B(X).
It is worth noticing that Definition 2.15 is well posed in the following sense.As soon as properties (RP.1) and (RP.2) are in force, if E ∈ B(X) has finite perimeter measure, then χ E ∈ L 0 (X, m) has finite variation measure with Var(χ E ; • ) = P(E; • ), since they are outer regular Borel measures on X agreeing on open sets.This is a simple consequence of Lemma 2.12.
By Definition 2.15, if u ∈ L 0 (X, m) has finite variation measure then, for each A ∈ B(X), we have Var(u; A) < +∞ and thus so that we can write In more general terms, we get the following extension of the relative coarea formula (2.6).Its proof follows from a routine approximation argument (see [9] for instance) and is thus omitted.

Corollary 2.16 (Generalized coarea formula
Keeping the same notation used in the previous sections, we let BV(X, m) := u ∈ L 1 (X, m) : u has finite variation measure .
Notice that, although BV(X, m) ⊂ BV (X, m) and BV (X, m) is a convex cone in L 1 (X, m), the set BV(X, m) may not be a convex cone in L 1 (X, m) as well, since the validity of the implication u, v ∈ BV(X, m) =⇒ u + v has finite variation measure is not automatically granted.For an example of such a phenomenon, we refer the interested reader to the variation of intrinsic maps between subgroups of sub-Riemannian Carnot groups [112,Rem. 4.2], but we will not enter into details of this issue because out of the scope of the present paper.
This being said, we introduce the following additional property for the relative perimeter P in (2.5) requiring the closure of BV(X, m) with respect to the sum of functions: We now outline some consequences of Lemma 2.13 and Proposition 2.14, and leave the simple proof of these statements to the interested reader, see also the proof of Lemma 2.10.

Chain rule.
We now establish a chain rule for the variation measure of continuous functions.To this aim, we need to assume the following locality property of the relative perimeter functional in (2.5): Loosely speaking, property (RP.L) states that, for any open set E ⊂ X, the relative perimeter functional A → P (E; A) is supported (in a measure-theoretic sense) on the topological boundary ∂E of the set E. Proof.Since ϕ is strictly increasing, its inverse function ϕ −1 : ϕ(R) → R is well defined, continuous and strictly increasing, and we can write Therefore, the set {ϕ(u) > t} has finite perimeter measure for L 1 -a.e.t ∈ R, with Hence, given A ∈ B(X), we have and so Performing a change of variables, we can write Now, since u ∈ C 0 (X), we know that {u > s} ∈ T and ∂{u > s} ⊂ {u = s} for all s ∈ R. Therefore, because of (RP.L), we have

By Corollary 2.16, we can write
so that, by combining all the above equalities, we conclude that for all A ∈ B(X), proving (2.8) and completing the proof.

p-slope and Sobolev functions. As customary, we let
If u ∈ W 1,1 (X, m), then we let |∇u| ∈ L 1 (X, m), |∇u| ≥ 0 m-a.e. in X, be the 1-slope of u, i.e., the unique L 1 (X, m) function such that From Corollary 2.17, we immediately deduce the following simple properties of 1-slopes of W 1,1 functions.

Definition 2.20 (p-relaxed 1-slope). Let p ∈ (1, +∞). We shall say that a function
Clearly, according to Definition 2.20 and thanks to the sequential compactness of weak topologies, if (2.9) Following the point of view of [10], one can prove the following basic properties of p-relaxed 1-slopes that will be useful in the sequel.

Lemma 2.21 (Basic properties of p-relaxed 1-slope). Let properties (RP.1), (RP.2), (RP.3), (RP.4) and (RP.+) be in force and let p ∈ (1, +∞). The following hold:
Under the assumptions of the above Lemma 2.21, for each u ∈ L p (X, m), the set Slope p (u) is a (possibly empty) closed convex subset of L p (X, m), and thus the following definition is well posed.

Definition 2.22 (Weak p-slope).
Let p ∈ (1, +∞) and let properties (RP.1), (RP.2), (RP.3), (RP.4) and (RP.+) be in force.If u ∈ L p (X, m) is such that Slope p (u) = ∅, we let |∇u| p be the element of Slope p (u) of minimal L p (X, m)norm and we call it the weak p-slope of u.Finally, we let Following the same line of [10], one can show that the weak p-slope can be actually approximated in L p (X, m) in the strong sense.

Cheeger sets in perimeter-measure spaces
In this section, we work in a measure space endowed with a perimeter functional as in Section 2.1.

3.1.
Existence of N-Cheeger sets.We begin by proving that the existence of N-Cheeger clusters of Ω is ensured whenever the perimeter functional possesses properties (P.4), (P.5), (P.6), and the set Ω ∈ A is N-admissible with finite m-measure.These requests though are not necessary, as some examples at the end of this section show.Theorem 3.1.Let properties (P.4), (P.5) and (P.6) be in force.Let N ∈ N, and let Ω ∈ A be a N-admissible set with m(Ω) ∈ (0, +∞).Then there exists a Cheeger N-cluster of Ω.
Proof.On the one hand, since Ω is N-admissible, there exists a N-cluster E ⊂ Ω, which immediately implies that h N (Ω) < +∞.On the other hand, for any N-cluster E = {E(i)} N i=1 of Ω, property (P.6) gives Clearly, for any k ∈ N sufficiently large and any i = 1, . . ., N, we have and thus sup which is finite, having assumed m(Ω) < +∞.By (P.4) and (P.5) (recall also Remark 2.1), possibly passing to a subsequence, for each i = 1, . . ., N, there exists as k → +∞.Now, using (P.6), for all k ∈ N sufficiently large and any i ∈ {1, . . ., N}, we get The behavior of f near zero prescribed by (P.6) immediately implies that m(E(i)) = 0 for all i ∈ {1, . . ., N}, as otherwise a contradiction would arise with h N (Ω) < +∞.Indeed, on the one hand being {E k (i)} k a minimizing sequence, and owing to (3.1), there exists k >> 1 such that for all k ≥ k we have f (m(E k (i))) ≤ 2h N (Ω).On the other hand, the isoperimetric property (P.6) implies there exists δ > 0 such that f (x) > 2h N (Ω) for all x ≤ δ.Hence, we deduce that m(E k (i)) ≥ δ for all i = 1, . . ., N and all k ≥ k.
It remains to be proved that E = {E(i)} N i=1 is a N-cluster contained in Ω is that the chambers E(i) are pairwise disjoint, and the reader can easily check it on its own.Consequently, thanks to (P.4), we find that and the conclusion follows.
Let us point out that properties (P.4), and (P.5), and (P.6) are all crucial in the above proof.Among them (P.6) looks as the "most artificial"; nevertheless it is essential in the reasoning: an example where existence fails when (P.6) is missing is given in Example 3.2 below.It is also relevant to point out that these properties provide a sufficient but in no way a necessary condition, as Example 3.
x > 1 , and P (•) is the Euclidean perimeter.In this setting, properties (P.1) through (P.5) hold, but (P.6) does not.Within this framework, one has h 1 (Ω) = 0, for any set Ω containing an open neighborhood of the origin.Indeed, it is enough to consider the sequence of balls centered at the origin B r ⊂ Ω (for r sufficiently small), for which we have Were to exist E ∈ C 1 (Ω), then P (E) = 0, and by the Euclidean isoperimetric inequality we would have |E| = 0. Being the weight w ∈ L 1 (R 2 ), this would eventually lead to contradicting the fact that m(E) > 0. This shows that Cheeger sets do not exist.In more generality, the same happens in any measure space (X, A , m) and in any set 1-admissible Ω ∈ A such that h 1 (Ω) = 0 and the only measurable subsets E of Ω with P (E) = 0 have zero m-measure.
For the sake of completeness, we shall note that, in the situation depicted in this remark, N-Cheeger sets exist in any open set Ω not containing the origin, since the weight w would be L ∞ (Ω), refer to [20,Prop. 3.3] or to [111,Prop. 3.2].
We now present two simple examples in which the existence of Cheeger sets is ensured even if properties (P.5) and (P.6) do not hold.
Example 3.3.Consider any non-negative (σ-finite) measure space (X, A , m), and consider as perimeter functional P (E) = m(E), for all E ∈ A .For this choice, while (P.4) holds, neither property (P.5) nor (P.6) hold, the latter because any isoperimetric function f is bounded from above by 1.Nevertheless, fixed any Ω ∈ A , we have h N (Ω) = N, for any integer N, and any N-cluster is a N-Cheeger set.

Inequalities between the Nand M-Cheeger constants.
Proposition 3.5.Let Ω ∈ A be a N-admissible set.Then, for all M ∈ N with M < N, one has Proof.Let M and N be fixed integers, with M < N. Let E be any fixed N-cluster.For any subset J M of {1, . . ., N} of cardinality M, the M-cluster {E(i)} i∈J M provides an upper bound to h M (Ω), while the (N − M)-cluster Hence, no matter how we choose J M , we have By taking the infimum among all N-clusters, the desired inequality follows.
Corollary 3.6.Let Ω ∈ A be a N-admissible set.Then, for all M ∈ N such that for some integer k one has N = kM, one has Remark 3.7.The inequalities in (3.2) and in (3.3) can be saturated, and this for instance happens anytime a set has multiple disjoint 1-Cheeger set.A trivial example of this behavior is given by N disjoint and equal balls in the usual Euclidean space.One can also build connected sets that have this feature.For N = 2, it is enough to consider a standard dumbbell in the usual 2-dimensional Euclidean space, that is, the set given by two disjoint equal balls, spaced sufficiently far apart, and connected via a thin tube.Such a set, has two connected 1-Cheeger sets E(1) and E(2) given by small perturbations of the two balls, and the 2-cluster E = {E(i)} is necessarily a 2-Cheeger set, refer for instance to [86,Ex. 4.5].
An easy connected example for N > 2 is instead given by a (N +2)-dumbbell in the usual 2-dimensional Euclidean space, that is, a set formed by N + 2 disjoint equal balls and linked by a thin tube, say For ε sufficiently small, and arguing as in [86,Ex. 4.5], it can be shown that such a set has N connected and disjoint 1-Cheeger sets, each corresponding to a small perturbation of the N balls with two neighboring ones.
3.3.M-subclusters of N-Cheeger set.Given a N-Cheeger set of Ω, consider any of its M-subcluster.It is natural to imagine that such a M-cluster is a M-Cheeger set in the ambient space given by Ω minus the N − M chambers not belonging to the subcluster.In this short section we prove that this is true.
For the sake of clarity of notation, we let |J| ∈ N ∪ {0} ∪ {+∞} be the cardinality of a set J ⊂ N.

Proposition 3.8. Let Ω ∈ A be a N-admissible set, and assume it has a
For any proper subset J ⊂ {1, . . ., N} let and let E J be the |J|-cluster given by Proof.It is enough to prove the claim for a subset J of cardinality N − 1, and then to reason by induction.In particular, up to relabeling, we can assume J to be the proper subset {1, . . ., N − 1}.
As both Ω and E(N) are measurable, so it is the set Ω J .Moreover, this latter is (N − 1)-admissible since there exists at least the (N − 1)-cluster m(E(i)) .
It is then immediate that the N-cluster (iii) if (P.4), (P.5), and (P.6) are in force, and Moreover, if also (P.3) is in force, P (Ω) is finite, and Proof.Recall that a N-admissible set Ω is also M-admissible for all integers M ≤ N, see Remark 2.5.Point (i).For any two fixed N-admissible sets with Ω 1 ⊂ Ω 2 , any M-cluster of Ω 1 is also a M-cluster of Ω 2 .The inequality immediately follows by definition of M-Cheeger constant.
Point (ii).In virtue of (3.3), and the positivity of h M (Ω) it is enough to prove the claim for M = 1.Fix ε > 0, and for all k, let C k ⊂ Ω k be such that Then by (P.6), we have and the claim follows by the monotonicity of the measure paired with the hypothesis that the m-measure of Ω k vanishes, that is, m(C k ) ≤ m(Ω k ) → 0, and the behavior of f prescribed by (P.6).Point (iii).Without loss of generality, we can assume there exists a constant as otherwise there is nothing to prove.Let us consider the (not relabeled) sequence realizing the lim inf.Since Ω k is converging L 1 (X, m) to a set of mfinite measure, we can also assume that m(Ω k ) is equibounded, independent of k, that is, Thus by Theorem 3.1 for each k there exists a M-Cheeger set E k for Ω k .Moreover, for any k we have Hence by (P.5), we can extract a (not relabeled) subsequence {E k (i)} k such that for all indexes i the chamber E k (i) converges in L 1 (X, m) to a limit set E(i) necessarily contained in Ω up to null sets.Moreover, by (P.6) one necessarily has m(E(i)) > 0 as otherwise a contradiction with the finiteness of lim inf k h M (Ω k ) would arise.Hence, E is a M-cluster for Ω.Thus, owing to (P.4) and the fact that E k is a M-Cheeger cluster of Ω k , we have that is, the first part of the claim.
To show the second part, let us pick a M-Cheeger cluster E for Ω, which exists since we are under the assumptions of Theorem 3.1.Let us consider the collections which, for k >> 1 are M-clusters for Ω k Clearly, for each fixed i we have that Therefore, by (P.3) for each i we have Taking the lim sup k , using the assumption of the convergence of P (Ω k ) we have Together with (P.4) this implies that, for each i, lim k P (E k (i)) exists and equals P (E(i)).Combining this fact, with the minimality of E for h M (Ω) and the first part of the claim, concludes the proof.
Remark 3.10.Notice that to prove point (iii) all requests come into play.Indeed, a-priori, one can work with an 'almost'-infimizing M-cluster for h M (Ω k ) and find an analogous of (3.5) up to an additive factor εm(Ω k ).Then, the compactness granted by (P.5) is needed, but in order to ensure that the limiting collection E is indeed a cluster, the isoperimetric property (P.6) is needed.Finally, when talking about a lim inf property, we cannot avoid enforcing (P.4).

Lemma 3.11.
Let Ω ∈ A be a N-admissible set with m(Ω) ∈ (0, +∞), and assume that C N (Ω) is not empty.If (P.6) is in force, then for every N-Cheeger set {C(i)} N i=1 ∈ C N (Ω), and for every i = 1, . . ., N, we have m(C(i)) ≥ c, where c > 0 is a constant depending only on h N (Ω) and the isoperimetric function f appearing in (P.6).Proof.Let {C(i)} N i=1 ∈ C N (Ω), then owing to (P.6), for every i we have By the assumptions on f given in (P.6), f (ε) → +∞ as ε → 0 + .Thus, there exists a threshold c = c(h Hence, for any N-Cheeger set, the lower bound on the volume of each of its chambers follows.
3.5.Additional properties of 1-Cheeger sets.For 1-Cheeger sets something more can be in general said, as we show in the next proposition.
Proposition 3.12.Let Ω ∈ A be a 1-admissible set, and assume that C 1 (Ω) is not empty.If (P.3) is in force, for any E, F ∈ C 1 (Ω), the following hold: Moreover, if also (P. 4) is in force, then C 1 (Ω) is closed with respect to countable unions and (m-non-negligible) intersections, that is, given any countable family {E j } j of 1-Cheeger sets, one has Proof.First, notice that we have the equalities Hence, the following chain of inequalities, owing to (P.3), holds and thus they are all equalities.This implies thus E ∪ F is a 1-Cheeger set, and so it is E ∩ F , provided that it has positive m-measure.This settles points (i) and (ii).Let now {E j } j be any countable family of 1-Cheeger sets.Then, the sequences are sequences of 1-Cheeger sets by points (i) and (ii) previously established (the second one, under the additional assumption that the intersections are m-non-negligible).Moreover, they converge, respectively, in L 1 (X, m) to the sets The lower-semicontinuity of P granted by (P.4) implies that these sets are 1-Cheeger sets themselves, and this concludes the proof.
Proof.Take a maximizing sequence The following uniform upper bound on the perimeters of {C k } k holds Thus by (P.5), up to extracting a subsequence, C k converge to some limit set C, which, by (P.4) is readily proven to be a 1-Cheeger set itself, provided that m(C) > 0, which holds true as we look for sets maximizing the L 1 (X, m) norm.
Assume now additionally (P.3), and let C 0 be a maximal 1-Cheeger set.For any other 1-Cheeger set C, if one were not to have C ⊂ C 0 a contradiction would immediately ensue, since C ∪ C 0 is itself a 1-Cheeger set by Proposition 3.12 (i).This same reasoning also yields the uniqueness of such a set.Example 3.14.An example of metric-measure space (X, A , m) where existence of maximal 1-Cheeger set fails is the following one.Consider (X, A , m) as the probability measure space (R 2 , B(R 2 ), (2π) − 1 2 e −̺ L 2 ), where ̺ = x , and consider as perimeter functional, the one for which P (E) = 0 for all E ∈ A excluded R 2 itself, and P (R 2 ) = 1.In such a setting neither (P.4) nor (P.5) hold.If we choose Ω = R 2 , all m-non-negligible sets are 1-Cheeger sets but for the whole space R 2 .No maximal 1-Cheeger set exists, as the supremum of their norms is 1, and this is the measure of the lone R 2 .
Proof.The proof is exactly the same as the one of Proposition 3.13, except that we now need to ensure that the limit set C is a viable competitor, that is, m(C) > 0. This is exactly why we need to require (P.6).The uniform lower bound on the volume of any 1-Cheeger set provided by Lemma 3.11, immediately allows to conclude.Example 3.16.An example of measure space (X, A , m) where existence of minimal 1-Cheeger set fails the one of Example 3.14.Chosen any Ω, all of its subsets but m-negligible ones are 1-Cheeger sets.Hence, minimal 1-Cheeger sets do not exists, being the infimum of the norms of 1-Cheeger sets zero.

Sets with prescribed mean curvature
In this section, we work in a measure space endowed with a perimeter functional as in Section 2.1.We show that the Cheeger constant acts as a threshold to determine whether non-trivial minimizers exist for a class of functionals, usually referred to as the prescribed mean curvature functional, under some suitable assumptions on the perimeter functional.In order to do so, the first thing we need to prove is the following lemma.Lemma 4.1.Let Ω ∈ A be a 1-admissible set.A m-non-negligible set C is a 1-Cheeger set of Ω if and only if it is a minimizer of Proof.It is sufficient to note that the inequality J h 1 (Ω) [F ] ≥ 0 holds true, and that the only m-non-negligible sets that can saturate it are 1-Cheeger sets.
The non-negativity of J h 1 (Ω) is trivial for m-negligible sets, while for all m-non-negligible sets, it follows by the definition of 1-Cheeger constant.
4.1.The variational mean curvature.Lemma 4.1 allows us to infer that any 1-Cheeger set has a P -mean curvature, that in the context of metricmeasure spaces becomes what is known as variational mean curvature, defined as follows.
Definition 4.2.Let (X, A , m) be a measure space, and let Ω ⊂ X be in among all m-measurable subsets F of Ω.
This definition was first given in measure spaces in [20] under some assumptions on the perimeter functional, and only when Ω = X, with the additional request m(X) < +∞.In the context of metric-measure spaces, one can give a notion of variational mean curvature, requiring the set E to be a local minimizer of the energy appearing in the previous definition, as follows.Definition 4.3.Let (X, d, m) be a metric-measure space, and let Ω ⊂ X be a m-measurable, open set.A function H ∈ L 1 (Ω, m) is said to be a variational mean curvature in Ω of a set E ⊂ Ω if E locally minimizes the functional that is, there exists a threshold r 0 > 0 such that for all r ≤ r 0 , for all x ∈ X and for all metric balls B r (x) ⋐ Ω, one has for all sets F ⊂ Ω with E∆F ⊂ B r (x).
In this context it becomes clearer why H is referred to as a variational mean curvature.Whenever the first variation of (4.2) is possible, one finds out that critical points have mean curvature given exactly by H.Alternatively, notice the following: whenever X is endowed with a topology, the class of measurable sets is that of Borel sets, and Ω is chosen as an open set, the defintion says that the set E, provided that H is in L ∞ (Ω, m), is a (Λ, r 0 )-minimizer of the perimeter.As soon as the perimeter functional is regular enough, the (Λ, r 0 )minimality implies that the set has constant mean curvature inside Ω given exactly by H.
These kind of curvatures have been studied for instance in [19] for the standard De Giorgi perimeter weighted with suitable density functions.
The following lemma is straightforward from the two definitions.
Lemma 4.4.Let (X, d, m) be a non-negative σ-finite metric-measure space, and let Ω ⊂ X be an open set.
Given Definition 4.2, a direct consequence of Lemma 4.1 is that 1-Cheeger sets of Ω, if they exist, have h 1 (Ω) as one of their variational mean curvatures in Ω -and this without assuming anything on the perimeter, apart from the non-negativity.Corollary 4.5.Let (X, A , m) be a non-negative σ-finite measure space and let Ω ∈ A be a 1-admissible set.
A similar result, limited to the P -mean curvature, can be inferred on each chamber of a N-Cheeger cluster, by simply using Proposition 3.8.Corollary 4.6.Let (X, A , m) be a non-negative σ-finite measure space and let Ω ∈ A be a N-admissible set.If E = {E(i)} N i=1 ∈ C N (Ω), then, letting J i = {i}, for every i = 1, . . ., N, h 1 (Ω J i ) is a P -mean curvature in Ω J i of the chamber E(i), where Ω J i is as in (3.4).
Proof.Let i be fixed.By Proposition 3.8, the chamber E(i) is a 1-Cheeger set of Ω J i .The conclusion now directly follows from Corollary 4.5.

4.2.
Relation with sets with prescribed P -mean curvature.Let now Ω ∈ A be fixed, and consider the functional that is, the same functional introduced in (4.1) but with a general positive constant κ ∈ R + in place of h 1 (Ω).
The reason why this functional is referred to as the prescribed P -mean curvature functional, is that the P -mean curvature of any minimizer E κ of the functional J κ in (4.4) is equal to κ.
The next theorem says that if there exists a 1-Cheeger set, and properties (P.1), (P.4), and (P.5) are in force, then (4.4) has m-non-negligible minimizers if and only κ ≥ h 1 (Ω).Theorem 4.7.Let (X, A , m) be a non-negative σ-finite measure space, and let Ω ∈ A with finite m-measure.For κ > 0, let J κ be the functional defined over m-measurable subsets of Ω.
Then, if properties (P.4), and (P.5) are in force, minimizers of J κ exist.In addition, if property (P.1) is also in force, the following hold true: Proof.First, we show that assumptions (P.4), and (P.5) and the finiteness of m(Ω) imply the existence of minimizers of J κ .Indeed, since the perimeter functional is non-negative, and by the monotonicity of measures, we have the trivial lower bound J κ ≥ −κm(Ω).Therefore, we can take an infimizing sequence {E k } k , for whose perimeters (for k large enough) we have the uniform upper bound which is finite by the finiteness of m(Ω).
By (P.5) we can extract a converging subsequence in L 1 (X, m) to some limit set E, and by (P.4) we have Hence, E is a minimizer of the problem.
Let us now turn our attention to points (i), (ii) and (iii).First, note that requiring (P.1) implies that the minimum of J κ is non-positive.Indeed, owing also to (P.4), Remark 2.1 gives that J κ is zero whenever evaluated on a mnegligible set.
Suppose that there exists a m-non-negligible minimizer E κ .Necessarily, by comparing with a m-negligible set we have and this latter ratio has to be greater than or equal to h 1 (Ω) by definition, thus point (i) is settled.
Conversely, let κ > h 1 (Ω), and let ε > 0 such that κ = h 1 (Ω) + 2ε.Since m(Ω) has finite measure, just as in the proof of Theorem 3.1, we can find an infimizing sequence {C j } j for the 1-Cheeger constant h 1 (Ω).For j >> 1, we have which yields the claim since J κ is zero when evaluated on m-negligible sets.This establishes point (ii).
Finally, assuming the existence of a 1-Cheeger set, point (iii) follows directly from (i), and (ii), and Lemma 4.1.

Relation with first 1-eigenvalue
In this section, we work in the setting introduced in Section 2.2, where we introduced BV functions starting from a given perimeter functional P on a measure space (X, A , m). (5.1) Here and in the following, we write u| X\Ω = 0 whenever X\Ω |u| dm = 0.
Analogously we can define the first 1-eigenvalue in the case of N-clusters as follows.
Definition 5.2 (First N-eigenvalue).Let properties (P.1) and (P.2) be in force.Let Ω ∈ A be a N-admissible set with m(Ω) ∈ (0, +∞).We define the first 1-eigenvalue for N-clusters relative to the variation on Ω as the quantity where the infimum is sought among the N-tuples for all i = j, i, j = 1, . . ., N.

5.2.
Relation with first 1-eigenvalue for N-clusters.We need the following preliminary result, which can be seen as a symmetric version of the coarea formula (2.3).Note that, in Lemma 5.3, we assume the validity of the symmetry property (P.7).
Proof.Equation ( 5.3) is a simple consequence of Cavalieri's principle, being where u + and u − are the positive and negative parts of u, respectively.To prove (5.4), we first observe that, by property (P.Therefore, we just need to show that P ({u ≤ t}) = P ({u < t}) for a.e.t < 0. Since clearly {u ≤ t} = {u < t} ∪ {u = t}, thanks to property (P.4) and Remark 2.1 it is enough to prove that m({u = t}) = 0 for a.e.t < 0. This is an immediate consequence of the fact that the function t → m({u ≤ t}) is monotone non-decreasing for t < 0, so that the set of its discontinuity points {t < 0 : m({u = t}) > 0} is at most countable.Now, the fact that χ F t ∈ BV (X, m) for a.e.t ∈ R directly follows from (5.4) and (5.3).Finally, if t > 0 then F t = {u > t} ⊂ Ω by definition of BV 0 (Ω, m).In a similar way, if t < 0, then The proof is complete.
We now assume the validity of properties (P.4) and (P.7), so that we can use Lemma 5.3, and focus on the lower bound for Λ N (Ω).
We begin with the case N = 1.By contradiction, start assuming that Let {F t : t ∈ R} be the family of sets introduced in Lemma 5.3 relatively to the function u.We claim that there exists t ∈ [0, +∞) such that m(F t ) > 0 either for all t > t or for all t < − t.
(5.6) Indeed, if either t 1 < t 2 < 0 or t 1 > t 2 > 0, then F t 1 ⊂ F t 2 by definition.Thus, it is enough to find t such that either m(F t) > 0 or m(F − t) > 0. If no such t exists, then Cavalieri's principle (5.3) implies that u 1 = 0, against our initial choice of u.Hence claim (5.6) follows and so, up to possibly replace u with −u, we can suppose that m(F t ) > 0 for t > t.Now, by Lemma 5.3, we can rewrite (5.5) as Recalling that Λ 1 (Ω) + ε ≤ h 1 (Ω) according to our initial assumption, from inequality (5.7) we immediately get that Therefore, we must have that for a.e.t ∈ R. Thus, taking into account that F t ⊂ Ω for all t ∈ R, we get that for a.e.t > t, that is, Λ 1 (Ω) + ε = h 1 (Ω) for all choices of ε > 0 suitably small, which is clearly impossible.Therefore, Λ 1 (Ω) ≥ h 1 Ω), as desired.
We now conclude the proof with the case N > 1.Let ε > 0 and let {u i : i = 1, . . ., N} be a viable N-tuple for the definition of Λ N such that For each i = 1, . . ., N, the function u i provides a viable competitor in the definition of Λ 1 .Consequently, using the inequality proved for the case N = 1, we get that The conclusion thus follows by letting ε → 0 + .
We end the present section with the following simple consequence of Theorem 5.4, and refer to [5,21] for the Euclidean case, and some final remarks.
Proof.Recalling that property (P.1) and (P.7) imply the validity of (P.2), we can argue as in the proof of Theorem 5.4, with the exception that we can now choose a minimizer u ∈ BV 0 (Ω, m) for Λ 1 (Ω), with no need to work with some given ε > 0. In particular, in place of (5.8), we obtain for all t ∈ R such that m(F t ) > 0, proving the first implication.On the other hand, owing again to Lemma 5.3, if all the level sets {F t : t ∈ R} of a viable competitor u ∈ BV 0 (Ω, m) with positive m-measure are 1-Cheeger sets, then so that u must be a minimizer attaining Λ 1 (Ω).Now, exploiting the first part of the claim, it is easy to show the part regarding uniqueness.First, notice that, given any 1-Cheeger set C, the function u = cχ C is a 1-eigenfunction for any c = 0. Thus, two different Cheeger sets must provide two different 1-eigenfunctions.Conversely, if 1 and u 2 are two distinct 1-eigenfunctions such that u 1 = cu 2 for all c = 0, then we can find t ∈ R such that the two level sets u 1 > t and u 2 > t have positive mmeasure and are distinct, hence identifying two different 1-Cheeger sets.
Remark 5.6.Let properties (P.1), (P.4) and (P.7) be in force.Whenever a 1-Cheeger set C exists (for instance under the assumptions of Theorem 3.1) Corollary 5.5 yields the existence of eigenfunctions of the variational problem defining Λ 1 (Ω), by setting u = cχ C , with c = 0. Remark 5.7.In Section 3.2 we already discussed some examples of sets Ω for which Nh 1 (Ω) = h N (Ω) (recall Remark 3.7).Thus, whenever Theorem 5.4 applies, we obtain that Λ N (Ω) equals these values for such sets Ω. Remark 5.8.We point out that, in [47,Thm. 3.1], the authors define Λ N (Ω) as the infimum of a different variational problem, staged on the skeleton of the ambient space X, and prove that it coincides with h N (Ω).To follow such an approach, one would need to require more properties on the ambient space X, at least starting with a vectorial structure.Remark 5.9.A key tool for the proof of Theorem 5.4 is the symmetric coarea formula of Lemma 5.3 which holds, provided that also (P.7) is enforced.In particular, this excludes anisotropic Euclidean settings where the Wulff shape is not central symmetric.A workaround, would be tweaking the variational problem defining Λ 1 , and rather define In such a way, in the proof of Theorem 5.4 the symmetric coarea formula would not be needed, and one could then establish the equality Whenever (P.7) holds as well, one gets the (more interesting) equality with the 1-eigenvalue Λ 1 (Ω).

Relation with first p-eigenvalue and p-torsion
In this section, we work in a topological non-negative σ-finite measure space with a relative perimeter functional as in Section 2.3.We discuss the relations between the 1-Cheeger constant and two other variational quantities, the first p-eigenvalue and the p-torsion function, extending to the present more general setting the results obtained in [82,83] and in [32].
In the following result, we prove that the 1-Cheeger constant h 1 (Ω) provides a lower bound on the first p-eigenvalue.
Assuming also (P.7), we can combine Theorem 6.3 with Theorem 5.4 obtaining the following corollary.Corollary 6.4.Let the assumptions of Theorem 6.3 be in force.If the perimeter in (2.7) also satisfies (P.7) and Ω ⊂ X is 1-admissible with respect to the variation in (2.7) with m(Ω) ∈ (0, +∞), then 6.2.Relation with p-torsional creep function.Assume properties (RP.1), (RP.2), (RP.3), (RP.4) and (RP.+) to be in force, let p ∈ (1, +∞), and let Ω ⊂ X be a non-empty p-regular open set with m(Ω) < +∞.We let be the p-torsional creep energy functional, see [84].When enough structure is on the ambient space, this is the Euler-Lagrange equation of the torsional creep PDE on ∂Ω, ( simply passing to its weak formulation, and writing down the first variation of the associated functional.Thanks to Lemma 2.21(i), it can be easily checked that the functional J p is strictly convex on the convex set W 1,p 0 (Ω, m).Hence, the torsional creep problem T p (Ω) := inf J p (u) : u ∈ W 1,p 0 (Ω, m) has at most one minimizer.If this exists, we denote it by w p ∈ W 1,p 0 (Ω, m).In particular, since 0 ∈ W 1,p 0 (Ω, m), we immediately see that Under the assumptions of Corollary 6.4, and assuming the existence of a non-trivial minimizer w p of J p , we can show that the 1-Cheeger constant of Ω provides a bound on the L 1 (X, m) norm of w p , in a similar fashion to [32].

Theorem 6.5 (Relation with the p-torsional creep function). Let the properties (RP.1), (RP.2), (RP.3), (RP.4), (RP.+), (RP.L), and (P.7) be in force and let p ∈ (1, +∞).
If Ω ⊂ X is a p-regular open set which is also 1admissible with respect to the variation in (2.7) with m(Ω) ∈ (0, +∞), and J p has a non-trivial minimizer w p = 0, then Proof.Using Corollary 6.4, the (non-trivial) torsional creep function w p as a competitor for λ 1,p (Ω), the inequality (6.4) and Hölder's inequality, we get Rearranging, the claimed inequality follows.Remark 6.6.If the weak formulation of the torsional creep PDE (6.3) is available, then one can test it against the solution w p itself, finding that the equality holds.Using this equality in place of inequality (6.4), one gets the analogous of (6.5) with a prefactor of p in place of p 1+ 1 p .Under additional structural hypotheses on the space, that allow to identify the p-slope with the p-th power of the 1-slope, one can altogether remove the prefactor from the inequality, similarly to [32, Thm.2].Remark 6.7.In the statement of Theorem 6.5, we need to assume that J p has a minimizer (which in case is the unique one, by strict convexity) and that this is non-trivial.This can be ensured in suitable spaces, where a Poincaré inequality holds, allowing to see that J p is coercive.

Applications
In this section we apply the general results presented above to specific settings.In each of the following examples we will consider the natural topology of the ambient space.7.1.Metric-measure spaces.Let (X, d) be a complete and separable metric space and let m be a non-negative Borel measure (on the σ-algebra induced by the distance d) which is finite on bounded Borel sets and such that supp m = X.In particular, m is a σ-finite measure on X.
Given u : X → R, we define the slope of u (also called the local Lipschitz constant of u) the function |∇u| : X → [0, +∞] defined as For an open set A ⊂ X, we say that u ∈ Lip loc (A) if for each x ∈ Ω there is r > 0 such that B r (x) ⊂ A and the restriction u| Br(x) is a Lipschitz function, where B r (x) ⊂ X denotes the d-ball centered at x ∈ X with radius r ∈ (0, +∞).
In the present metric-measure setting, one has the following natural definition of BV functions, see [12,13,103].Definition 7.1 (BV functions in metric spaces).We say that u ∈ BV (X, d, m) if u ∈ L 1 (X, m) and there exists a sequence (u k ) k∈N ⊂ Lip loc (X) such that Moreover, we let restriction to open sets of a finite Borel measure, for which we keep the same notation.This result was originally proved in [103] for locally compact metric spaces, and then generalized to the possibly non-locally compact setting in [12].Actually, as done in [103], the convergence in L 1 (A, m) in (7.1) may be replaced with the convergence in L 1 loc (A, m) without affecting the overall approach.In the present setting, the (total) perimeter functional given by (7.1) in Definition 7.1 satisfies the properties (P.1), (P.2), (P.3), (P.4), (P.5) and (P.7).Indeed, properties (P.1), (P.2) and (P.7) immediately follow from Definition 7.1.For property (P.3), we refer to [103,Prop. 4.7(3)].Finally, property (P.4) is a consequence of [103,Prop. 3.6] and property (P.5) follows from [103,Thm. 3.7].
For what concerns the variation measure introduced in (7.In virtue of the properties listed above, we easily deduce the validity of the relation between the 1-Cheeger constant of an open set Ω ⊂ X with m(Ω) ∈ (0, +∞) and the first 1-eigenvalue as in Theorem 5.4, meaning that Incidentally, we refer the reader to [12,Sect. 6] for the definition of the 1-Laplacian operator in this general context.
In the metric-measure framework, the definition of W 1,1 (X, d, m) is not completely understood, see the discussion in [12,Sect. 8].As usual, one possibility is to say that u ∈ W 1,1 (X, d, m) if u ∈ BV (X, d, m) and |Du| ≪ m, and then to proceed as in Section 2.3.3 in order to work out the machinery needed to establish the relation with between the 1-Cheeger constant and the first peigenvalue in Corollary 6.4.However, one can exploit the richer structure of the ambient space to get a more direct and plainer approach to the relation with the first p-eigenvalue.
Let us briefly detail the overall idea.In the spirit of [60] and in an analogous way of Definition 7.1 (see the discussion at the end of [103, Sect.2]), we say that u ∈ W 1,p (X, d, m) for some p ∈ (1, +∞) if there exists a sequence (u k ) k∈N ⊂ Lip loc (X) such that Therefore, we can consider the Cheeger p-energy of u, defined by 3) as the natural replacement of the Dirichlet p-energy in this framework.
Accordingly, for a given non-empty open set Ω ⊂ X, we say that u ∈ W 1,p 0 (Ω, d, m) if there exists a sequence (u k ) k∈N ⊂ Lip loc (X) as in (7.2) (so that, in particular, u ∈ W 1,p (X, d, m)) with the additional property that supp u k ⋐ Ω for all k ∈ N. Therefore, coherently with what was done in Definition 6.2, we let Now, it is not difficult to see that, in virtue of the definition of the Cheeger p-energy in (7.3), the infimum in (7.4) can be actually restricted to functions u ∈ L p (X, m) ∩ Lip loc (X) such that Consequently, by the definition in (7.1) and Hölder's inequality, we can estimate Therefore and thus d, m) p p by the arbitrariness of u in the right-hand side of (7.6), proving Corollary 6.4.Similar considerations can be done for the relation of the 1-Cheeger constant with the p-torsional creep function as in Theorem 6.5.We leave the analogous details to the interested reader.

Euclidean spaces with density.
Let A = B(R n ) be the Borel σalgebra in R n and consider two lower-semicontinuous density functions g ∈ L ∞ (R n ; [0, +∞)), and h ∈ L ∞ (R n × S n−1 ; (0, +∞)), so that h is convex in the second variable and locally bounded away from zero, i.e., for any bounded set Ω ⊂ R n there exists C > 0 such that For any E ∈ A , we let the weighted volume and perimeter of E to be defined respectively by ) Here, we denote by ∂ * E the reduced boundary of E and, for every x ∈ ∂ * E, ν E (x) the outer unit normal vector to E at x (see [9] for definitions and properties of sets of finite perimeter).
If g and h are identically equal to 1, m g (•) = L n is the Lebesgue measure, and P h = P Eucl is the standard Euclidean perimeter that satisfies all properties (P.1)-(P.6)and (P.7) in the measure space (R n , A , m).In particular, (P.6) holds with f (ε) = nω 1/n n ε −1/n , and follows from the standard isoperimetric inequality P Eucl (E) ≥ nω 1/n n L n (E) 1− 1 n , (7.10) holding for any E ∈ A .The Cheeger problem in this setting is standard, see [87,108] and its minimizers are now completely characterized for a large class of planar sets [38,82,88,90,111], and reasonably well understood for convex N-dimensional bodies [5,26].Recently, Cheeger clusters have been introduced and studied in [46], see also [30,31,47].Interestingly, the Euclidean Cheeger problem plays a role in the ROF model for regularization of noisy images, as highlighted in [5], see also [87,Sect. 2.3], and this is also linked to our Section 4 and to our Corollary 5.5.
We now turn to the case of volume and perimeter with densities, for which the Cheeger problem has been considered for N = 1, e.g., in [49,79,83,91,111].We discuss properties (P.1)-(P.7)for the general densities h, g above.Notice that (7.7) implies that sets with locally finite perimeter P h are all and only those with locally finite Euclidean perimeter.Properties (P.1), (P.2) are immediate from the definitions.Given E, F ∈ A , (P.3) follows from the validity of the following relations between reduced boundaries and sets operations: Here, E (t) denotes the set of points of density t ≥ 0 for E and we recall that L n (R n \(E (0) ∪E (1) )) = 0, see, e.g., [93,Thm. 16.3] for more details.Property (P.4) holds true thanks to Reshetnyak lower-semicontinuity theorem, see [9,Thm. 2.38] and uses the lower-semicontinuity and the convexity assumptions on h.Property (P.5) follows from the standard compactness theorem for sets with equibounded perimeter and from in (7.7).Property (P.7) is equivalent to assume that h is even in the second variable.
Finally, we discuss (P.6).If (7.7) also holds on Ω = R n , the isoperimetric inequality (7.10) extends to this setting by observing that for any ε > 0 and E ∈ A such that m(A) ≤ ε we have with f (ε In this case, Theorem 3.1 implies existence of Cheeger N-clusters of any admissible set Ω ⊂ R n for perimeter and volume with double (anisotropic) densities.Observe that, in order for Ω to be admissible, it is needed that g > 0 on a Borel set of positive measure contained in Ω.This covers the existence results already present in the literature for N = 1 [49,83,111] and double densities or for Euclidean densities and N > 1 [46,47], and generalize it to the case of double density and N > 1.
In the general case where (7.7) does not extend to a global bound, (P.6) might not hold.Some specific examples of this type are discussed in the following subsections.Nonetheless, Theorem 4.7 applies to general densities, establishing the relation between the 1-Cheeger constant with the curvature functional, as previously discussed, e.g., in [8,49].Assuming the symmetry assumption (P.7), Theorem 5.4 shows that the 1-Cheeger constant h 1 (Ω) corresponds to the first 1-eigenvalue λ 1,1 (Ω) defined in Definition 5.1.
Estimates of the first p-eigenvalue in terms of the 1-Cheeger constant are proved in [83] for anisotropic (symmetric) perimeters whose density does not depend on the position, and Euclidean volume.In this setting, we observe that P h admits the following distributional formulation where h * denotes the dual norm to h.The latter yields to the validity of (RP.L) and (RP.+), thus allowing to apply Theorem 6.3 and establish a relation between the 1-Cheeger constant and the first p-eigenvalue defined in Definition 6.2 in the same spirit of [83].As far as we know the relation between the 1-Cheeger constant for more general densities h = h(x, ν) and the spectrum of specific "p-Laplace" operators in the spirit of [83] is an open question.
7.2.1.Gauss space.When (7.7) does not extend to Ω = R n , property (P.6) cannot be deduced as in (7.11).A specific setting where this happens is the Gauss space, corresponding to the choice g = h = γ, where 2 .(7.12) Let us notice that (7.7)only holds locally and that (R n , m γ ) is a probability space, i.e., Similarly to the Euclidean case, we can define the Gaussian perimeter of a Borel set E inside an open set Ω ⊂ R n as 13) It is easy to see that if a set E has finite Gaussian perimeter then it also has locally finite Euclidean perimeter and Properties (P.1)-(P.5),and (P.7) follow as above.We discuss the isoperimetric property (P.6).For every Borel set E ⊂ R n , the following isoperimetric inequality holds (see [18,27,116]) where U : R → R is the isoperimetric function defined as and it has the following asymptotic behavior [50] We deduce (P.6) by setting f : (0, +∞) → (0, +∞) as Moreover, using the distributional formulation (7.13), one can deduce the validity of (RP.+), (RP.L).Therefore, all of our results apply in this setting.While the existence of 1-Cheeger sets was already known, see [50,81], and the clustering isoperimetric problem has been recently addressed in [101], the cluster problem for N > 1 had never been treated.The relation with the prescribed curvature functional had been studied in [50], while, up to our knowledge, the relation with the first eigenvalue of the Dirichlet p-Laplacian had never been proved, but only quickly observed in [81] for p = 2 (the Ornstein-Uhlenbeck operator).

Monomial and radial weights.
Further settings where (P.6) cannot be deduced from (7.7), are those of monomial and radial densities.Given A = (a 1 , . . ., a n ) ∈ R n , so that a i ≥ 0, for i = 1, . . ., n, a monomial weight is g(x) = x A , where we used the notation A radial weight, instead, is of the type g(x) = x q , q ≥ 0.
The validity of (P. 5) is an open problem in general.Some hypotheses on K granting the compactness are currently being studied in the forthcoming paper [25]; assuming them, one can apply Theorem 3.1 and Theorem 4.7.
An explicit example in which (7.17), (7.19), (7.20) hold true, and the validity of (P.5) is proved (at least suitably restricting the ambient measure space) is given by the fractional s-perimeter, corresponding to the choice K s (x) = |x| −n−s .In this case, we let ω be a bounded extension domain for the fractional Sobolev space W s,p , see definitions in [69].Then, on the measure space (ω, B(ω), L n ( • )), P s satisfies (P.5) by [69,Thm. 7.1].For a reference on non-local perimeter functionals we refer to [97], and specifically to [97,Sect. 6] for some results on non-local Cheeger problems.We mention that the Cheeger problem for P s and the Lebesgue measure m = L n ( • ) has been introduced and studied in [28], where existence of N-Cheeger sets of a bounded open set Ω ⊂ R n is proved for N = 1.Up to our knowledge, for N > 1 the N-Cheeger problem has never been considered in this setting, while the clustering isoperimetric problem has been treated in [56].We refer to the recent paper [24, Thm.1.5] for a discussion of the fractional Cheeger constant with respect to the existence of minimizers of the prescribed mean curvature functional.
7.3.2.Distributional fractional perimeters.In [62], a new space BV s (R n ) of functions with bounded fractional variation on R n of order s ∈ (0, 1) is introduced via a distributional approach exploiting suitable notions of fractional gradient and divergence.More precisely, the fractional s-variation of a function u ∈ L 1 (R n ) is defined as and µ n,s is a normalization constant.The distributional fractional s-perimeter of a Lebesgue measurable set E ⊂ R n is then defined as the total fractional variation of its characteristic function |D s χ E |(R n ).In [62], the authors show that |D s χ E |(R n ) ≤ µ n,s P s (E) whenever E is a measurable set, where P s is as in (7.21), thus showing that the distributional approach allows to enlarge the usual notion of fractional perimeter.
Following [62], the functional E → |D s χ E |(R n ) enjoys several properties on measurable sets of R n .In fact, (P.1), (P.2) are direct consequences of the definition, yielding the validity of the first part of Theorem 5.4.Moreover, properties (P.4), (P.5) and (P.6) are respectively proved in [62,Prop. 4.3], [62,Thm. 3.16], and [62,Thm. 4.4] (provided that n ≥ 2, see [63,Thm. 3.8] for the case n = 1).In addition, property (P.7) trivially follows from the definition.In particular, we are in a position to apply Theorem 3.1 and Theorem 4.7.Finally, the validity of (P. 3) is open, while it is known that (RP.L) is false in general [62,Rem. 4.9], thus we can not apply the results concerning the first eigenvalue of the general Dirichlet p-Laplacian (it is worth noticing that the local form of the chain rule in this context is false [64], so a direct adaptation of the proof of Theorem 6.3 in this framework is not clear).
We remark that the aforementioned results have never been proved before in this specific non-local setting.7.3.3.Fractional Gauss spaces.For x, y ∈ R n , t > 0, let M t (x, y) be the Mehler kernel (see [43][44][45] for the precise definition), for any σ > 0 we set dt, where the integral has to be intended in the Cauchy principal value sense.
For A, B measurable and disjoint subsets of R n , we denote with L s (A, B) the (s-Gaussian) interaction functional, i.e., where γ is as in (7.12).Let Ω ⊂ R n be a connected open set with Lipschitz boundary, and E ⊂ R n be a measurable set.The Gaussian s-perimeter of E in Ω is defined as follows Properties (P.1) and (P.2) follow directly from the definition, whereas (P.7) is a consequence of the symmetry of the kernel K s (x, y), for (P.5) and (P.4) we refer to [43][44][45].Therefore Theorem 4.7 and Theorem 5.4 hold true.Property (P.3) can be proved exactly as in [55, Prop.2.2] Concerning (P.6), the following isoperimetric inequality is proved for every Borel set E ⊂ R n in [106] P s (E) ≥ I s (γ(E)) (7.22)where I : (0, 1) → (0, +∞) is the fractional Gaussian isoperimetric function, i.e., the function that associates to any m in (0, 1) the fractional Gaussian perimeter of a halfspace having Gaussian measure m.As far as we know the asymptotic behavior of I s (m) as m → 0 + is not known, for this reason we can not guarantee the validity of (P.6).
7.4.Riemannian manifolds.Let (M, g) be a complete Riemannian manifold of dimension n ∈ N. When endowed with its distance, it is a separable metric space and its volume measure is a a non-negative Borel measure which is finite on bounded Borel sets, so that one can rely on the discussion made in Section 7.1 to obtain the validity of (P.1), (P.2), (P.3), (P.4), (P.5) and (P.7).On the other hand, if M has non-negative Ricci curvature, then property (P.6) is a consequence of the sharp isoperimetric inequality recently obtained by Brendle in [29] for non-compact manifolds with Euclidean volume growth, see also [3].If M is compact, this is in the same spirit of the celebrated Lévy-Gromov isoperimetric inequality, see [77,App. C].If the Ricci curvature bound is negative, no isoperimetric inequality can be derived in general without further assumptions on the manifold itself, such as lower bounds on the diameter of M, see [100] and references therein for a more detailed discussion.Following the strategy presented in Section 7.1, the results contained in our paper then allow to recover Cheeger inequalities in Riemannian manifolds with non-negative curvature, in the spirit of the original appearance of Cheeger inequalities in compact Riemannian manifolds, due to Cheeger [59] for p = 2.The results of our paper also cover the existence of Cheeger sets, originally proved in [36] for compact Riemannian manifolds (see also [23]), and the links with the the prescribed mean curvature.We refer to [41] for the relation between the Cheeger constant and the torsion problem (6.3) for p = 2 in compact Riemannian manifolds.7.5.CD-spaces.CD-spaces are metric-measure spaces generalizing Riemannian manifolds with Ricci curvature bounded from below, via assumptions on a synthetic notion of curvature, encoded in the so-called curvature-dimension condition CD(K, n) for K ∈ R and n ≥ 1, see the cornerstones [92,114,115].Geometric Analysis on these non-smooth spaces is subject to a great interest in the recent years, see e.g., [10,11,51,52].
As CD(K, n) for ∈ R and n ≥ 1 are complete metric spaces endowed with a Borel measure m which is finite on bounded Borel sets, the discussion made in Section 7.1 applies to this framework, yielding the validity of properties (P.1), (P.2), (P.3), (P.4), (P.5) and (P.7).Concerning (P.6), for K ≥ 0, a sharp isoperimetric inequality has been recently proved in [15] for the subclass of RCD(0, n)-spaces with Euclidean volume growth, when m = H n , yielding (P.6) with f (ε) = nω 1/n n AVR(X) 1/n ε −1/n .Here AVR(X) stands for the asymptotic volume ratio, assumed to be in (0, 1].We also refer to [14,Thm. 3.19 and Rem. 3.20] where the validity of property (P.6) is discussed in more general metric-measure spaces with particular attention to the case of CD(K, n) spaces for K ∈ R and 1 < n < +∞, and to the celebrated Lévy-Gromov isoperimetric inequality proved in [52,54] holding for essentially non branching CD(K, n) spaces with finite diameter.
When (P.6) holds true, all the results contained in our paper then apply, establishing existence of Cheeger sets, relations with the the prescribed curvature functional, and Cheeger inequalities following the strategy presented in Section 7.1.The equivalence of the Cheeger constant and the first 1-eigenvalue of the Laplacian was previously pointed out in [53,Section 5] for more general metric-measure spaces including non branching CD(K, n) spaces.Lower bounds on the Cheeger constant for (essentially non-branching) CD * (K, n) spaces are proved in [52][53][54].7.6.Carnot-Carathéodory spaces.Let ω ⊂ R n be open and connected and let X = {X 1 , . . ., X k } be vector fields in ω with real C ∞ -smooth coefficients.An absolutely continuous curve γ : [0, T ] → ω is said to be admissible if there exists u = (u 1 , . . ., u k ) ∈ L 1 ([0, T ]) so that Given two points x, y ∈ ω we let d cc (x, y) the Carnot-Carathéodory distance between x and y be defined as the shortest length of admissible curves connecting them.We assume that the Hörmander condition rank(Lie X ) = n on the Lie algebra Lie X generated by X holds true.Then d cc (x, y) < ∞ for any couple of points x, y ∈ ω thanks to Chow-Rashewski Theorem, see [4] for details.The metric space (ω, d cc ) is called a Carnot-Carathéodory space, and it is separable.Assuming (ω, d cc ) to be also complete and endowing it with the Lebesgue measure L n , one is then allowed to rely on the discussion of Section 7.1 to guarantee the validity of properties (P.1), (P.2), (P.3), (P.4), (P.5) and (P.7) for the distributional perimeter of Definition 7.1.One can see that this actually corresponds to the so-called X -perimeter, introduced in [42] and then systematically studied in [71,72].
We discuss the validity of (P.6).We first observe that, as summarized in [78,Sect. 11.4], up to taking a smaller ω, we are ensured (globally in ω) the validity of a doubling property for metric balls and of a (1, 1)-Poincaré inequality for the horizontal gradient ∇ X u = k i=1 X i uX i , thanks to the celebrated works by Nagel, Stein, and Wainger [105] and Jerison [80] respectively.This allows to rely on the results of [72,Thm. 1.18], guaranteeing the validity of the following isoperimetric inequality for any Lebesgue measurable set E ⊂ ω Here C X is a positive constant depending on ω and X , and Q ≥ n is the so called homogeneous dimension.Property (P.6) then follows with f (ε) = C X ε −1/Q .
All the results contained in our paper then apply to this setting, establishing existence of Cheeger sets (Theorem 3.1), relations with the the prescribed curvature functional (Theorem 4.7), and Cheeger inequalities following the strategy presented in Section 7.1.We observe that, following [70,Cor. 11], and recalling that the topology induced by d cc is equivalent to the Euclidean one [4,Thm. 3.31], the Sobolev space W 1,p (ω, L n ( • )) introduced in Definition 2.22 satisfies the following: where ∆ X u = k i=1 X 2 i u is the so-called hypoelliptic sub-Laplacian associated with X .In particular, (6.2) gives a lower bound for the bottom of the spectrum of −∆ X on Ω. Cheeger's inequalities of this type have already been investigated in [104] in the context of Carnot groups.This paper extends them to more general Carnot-Carathéodory structures.

Metric graphs.
A graph G is a couple (V, E) consisting of a finite or countable set of vertices V = {v i } and a prescribed set of edges E = {e j } connecting the vertices.For simplicity, we assume the absence of loops and of multiple edges.

Definition 2 . 6
(N-Cheeger constant and N-Cheeger sets).Let N ∈ N and let Ω ∈ A be a N-admissible set.The N-Cheeger constant of Ω is m) has finite variation measure, then u + c has finite variation measure, with |D(u + c)| = |Du|, for all c ∈ R; (iii) constant functions have finite variation measure and |Dc| = 0 for all

Example 3 . 2 .
3 and Example 3.4 show.Consider the measure space

3. 4 .Proposition 3 . 9 (
Properties of N-Cheeger sets.Basic properties of N-Cheeger sets).Let {Ω k } k ⊂ A be a collection of N-admissible set.The following hold for all integers M ≤ N: