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Mathematics, Functional Analysis, math.FA,
Abstract:
We study a modified version of Lerman-Whitehouse Menger-like curvature
defined for m+2 points in an n-dimensional Euclidean space. For 1 <= l <= m+2
and an m-dimensional subset S of R^n we also introduce global versions of this
discrete curvature, by taking supremum with respect to m+2-l points on S. We
then define geometric curvature energies by integrating one of the global
Menger-like curvatures, raised to a certain power p, over all l-tuples of
points on S. Next, we prove that if S is compact and m-Ahlfors regular and if p
is greater than ml, then the P. Jones' \beta-numbers of S must decay as r^t
with r \to 0 for some t in (0,1). If S is an immersed C^1 manifold or a
bilipschitz image of such set then it follows that it is Reifenberg flat with
vanishing constant, hence (by a theorem of David, Kenig and Toro) an embedded
C^{1,t} manifold. We also define a wide class of other sets for which this
assertion is true. After that, we bootstrap the exponent t to the optimal one a
= 1 - ml/p showing an analogue of the Morrey-Sobolev embedding theorem.
Moreover, we obtain a qualitative control over the local graph representations
of S only in terms of the energy.