Integral Menger curvature for sets of arbitrary dimension and codimension

Abstract

We propose a notion of integral Menger curvature for compact, $m$-dimensional sets in $n$-dimensional Euclidean space and prove that finiteness of this quantity implies that the set is $C^{1,\alpha }$ embedded manifold with the H{ö}lder norm and the size of maps depending only on the curvature. We develop the ideas introduced by Strzelecki and von der Mosel [Adv. Math. 226(2011)] and use a similar strategy to prove our results.

This dissertation is not going to be published. The article "Geometric Sobolev-like embedding using high-dimensional Menger-like curvature" [arXiv:1205.4112] obsoletes my thesis. For any further reference one should use [arXiv:1205.4112] or its published version

Cited by in corpus (6)

• On some knot energies involving Menger curvature
• Characterizing $W^{2,p}$~submanifolds by $p$-integrability of global curvatures
• Geometric Sobolev-like embedding using high-dimensional Menger-like curvature
• Sharp Boundedness and Regularizing effects of the integral Menger curvature for submanifolds
• Minimal Hölder regularity implying finiteness of integral Menger curvature
• M{ö}bius-invariant self-avoidance energies for non-smooth sets in arbitrary dimensions