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

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