Tangency properties of sets with finite geometric curvature energies

TL;DR

This paper explores how finite geometric curvature energies influence the existence of approximate tangents in sets, establishing regularity results, sharpness of exponents, and optimality of conditions.

Contribution

It demonstrates new regularity properties of sets with finite curvature energies, identifies sharp thresholds for exponents, and provides examples showing the limits of these properties.

Findings

Finite inverse thickness implies approximate $eta$-tangent existence.

Finite integral Menger curvature energies imply weak approximate $eta$-tangent existence.

Sharpness of exponents for $eta=1$, with examples showing limits of regularity.

Abstract

We investigate inverse thickness $1/\Delta$ and the integral Menger curvature energies $\mathcal{U}_{p}^{\alpha}$, $\mathcal{I}_{p}^{\alpha}$ and $\mathcal{M}_{p}^{\alpha}$, to find that finite $1/\Delta$ or $\mathcal{U}_{p}^{\alpha}$ implies the existence of an approximate $\alpha$-tangent at all points of the set, when $p\geq \alpha$ and that finite $\mathcal{I}_{p}^{\alpha}$ or $\mathcal{M}_{p}^{\alpha}$ implies the existence of a weak approximate $\alpha$-tangent at every point of the set for $p\geq 2\alpha$ or $p\geq 3\alpha$, respectively, if some additional density properties hold. This includes the scale invariant case $p=2$ for $\mathcal{I}_{p}^{1}$ and $p=3$ for $\mathcal{M}_{p}^{1}$, for which, to the best of our knowledge, no regularity properties are established up to now. Furthermore we prove that for $\alpha=1$ these exponents are sharp, i.e., that if $p$ lies below the threshold value of scale innvariance, then there exists a set containing points with no (weak) approximate 1-tangent, but such that the corresponding energy is still finite. For $\mathcal{I}_{p}^{1}$ and $\mathcal{M}_{p}^{1}$ we give an example of a set which possesses a point that has no approximate 1-tangent, but finite energy for all $p\in (0,\infty)$ and thus show that the existence of weak approximate 1-tangents is the most we can expect, in other words our results are also optimal in this respect.