Tangent-point repulsive potentials for a class of non-smooth $m$-dimensional sets in ${\mathbb{R}}^n$. Part I: Smoothing and self-avoidance effects

We consider repulsive potential energies ${\text{E}}_q(\mathrm{\Sigma })$, whose integrand measures tangent-point interactions, on a large class of non-smooth $m$-dimensional sets $\mathrm{\Sigma }$ in ${\mathbb{R}}^n\mathrm{.}$ Finiteness of the energy ${\text{E}}_q(\mathrm{\Sigma })$ has three sorts of effects for the set $\mathrm{\Sigma }$: topological effects excluding all kinds of (a priori admissible) self-intersections, geometric and measure-theoretic effects, providing large projections of $\mathrm{\Sigma }$ onto suitable $m$-planes and therefore large $m$-dimensional Hausdorff measure of $\mathrm{\Sigma }$ within small balls up to a uniformly controlled scale, and finally, regularizing effects culminating in a geometric variant of the Morrey-Sobolev embedding theorem: Any admissible set $\mathrm{\Sigma }$ with finite ${\text{E}}_q$-energy, for any exponent $q>2m$, is, in fact, a $C^1$-manifold whose tangent planes vary in a H"older continuous manner with the optimal H"older exponent $\mu =1-(2m)\mathrm{/}q$. Moreover, the patch size of the local $C^{1,\mu }$-graph representations is uniformly controlled from below only in terms of the energy value ${\text{E}}_q(\mathrm{\Sigma })$.