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A note on integral Menger curvature for curves



For continuously differentiable embedded curves γ, we will give a necessary and sufficient condition for the boundedness of integral versions of the Menger curvature. We will show that for p > 3 the integral Menger curvature equation image is finite if and only if γ belongs to the Sobolev Slobodeckij space equation image. The quantity equation image – defined by taking the supremum of the p-th power of Menger curvature with respect to one variable and then integrating over the remaining two – is finite for p > 2, if and only if γ belongs to the space equation image.