Tangency properties of sets with finite geometric curvature energies

Sebastian Scholtes¹

¹ Institut für Mathematik RWTH Aachen University Templergraben 55 D-52062 Aachen, Germany

Fundamenta Mathematicae, Tome 218 (2012) no. 2, pp. 165-191

Résumé

We investigate tangential regularity properties of sets of fractal dimension, whose inverse thickness or integral Menger curvature energies are bounded. For the most prominent of these energies, the integral Menger curvature \begin{equation*} \newcommand{\dHM}{\,d\mathcal H^{\he}}\newcommand{\he}{{\alpha}} \mathcal{M}_{p}^{\he}(X):=\int_{X}\int_{X}\int_{X}\kappa^{p}(x,y,z)\dHM_{X}(x)\dHM_{X}(y)\dHM_{X}(z), \end{equation*} where $\kappa(x,y,z)$ is the inverse circumradius of the triangle defined by $x,y$ and $z$, we find that $\mathcal M_{p}^{\he}(X)\infty$ for $p\geq 3\he$ implies the existence of a weak approximate $\newcommand{\he}{{\alpha}}\he$-tangent at every point of the set, if some mild density properties hold. This includes the scale invariant case $p=3$ for $\mathcal{M}_{p}^{1}$, for which, to the best of our knowledge, no regularity properties have been established before. Furthermore we prove that for $\he=1$ these exponents are sharp, i.e., if $p$ lies below the threshold value of scale invariance, then there exists a set containing points with no weak approximate $1$-tangent, but such that the energy is still finite. Moreover we demonstrate that weak approximate tangents are the most we can expect. For the other curvature energies analogous results are shown.

Détail

DOI : 10.4064/fm218-2-4

Keywords: investigate tangential regularity properties sets fractal dimension whose inverse thickness integral menger curvature energies bounded prominent these energies integral menger curvature begin equation* newcommand dhm mathcal newcommand alpha mathcal int int int kappa dhm dhm dhm end equation* where kappa inverse circumradius triangle defined mathcal infty geq implies existence weak approximate newcommand alpha he tangent every point set mild density properties includes scale invariant mathcal which best knowledge regularity properties have established before furthermore prove these exponents sharp lies below threshold value scale invariance there exists set containing points weak approximate tangent energy still finite moreover demonstrate weak approximate tangents expect other curvature energies analogous results shown