Tangency properties of sets with finite geometric curvature energies
Tom 218 / 2012
Streszczenie
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.
