Fine properties of the curvature of arbitrary closed sets

Abstract

Given an arbitrary closed set A of ${\mathbf{R}}^n$, we establish the relation between the eigenvalues of the approximate differential of the spherical image map of A and the principal curvatures of A introduced by Hug-Last-Weil, thus extending a well known relation for sets of positive reach by Federer and Zaehle. Then we provide for every $m=1,\dots ,n-1$ an integral representation for the support measure ${\mu }_m$ of A with respect to the m dimensional Hausdoff measure. Moreover a notion of second fundamental form $Q_A$ for an arbitrary closed set A is introduced so that the finite principal curvatures of A correspond to the eigenvalues of $Q_A$. We prove that the approximate differential of order 2, introduced in a previous work of the author, equals in a certain sense the absolutely continuous part of $Q_A$, thus providing a natural generalization to higher order differentiability of the classical result of Calderon and Zygmund on the approximate differentiability of functions of bounded variation.

27 pages. This preprint expands sections 2-5 of version v1 of this submission. Sections 6-7 of v1 will be moved in seperate pre-prints

References in corpus (1)

• Rectifiability and approximate differentiability of higher order for sets

Cited by in corpus (3)

• Uniqueness of critical points of the anisotropic isoperimetric problem for finite perimeter sets
• Second order rectifiability of varifolds of bounded mean curvature
• $C^{1,1}$-rectifiability and Heintze-Karcher inequality on ${\mathbf{S}}^{n+1}$