Tangency properties of sets with finite geometric curvature energies

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 $p\ge \alpha$ 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 $p\ge 2\alpha$ or $p\ge 3\alpha$, 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.

arXiv comment38 pages