Fine properties of the curvature of arbitrary closed sets

TL;DR

This paper extends classical curvature relations to arbitrary closed sets in Euclidean space, introduces a second fundamental form for such sets, and provides integral representations for support measures, broadening geometric analysis tools.

Contribution

It generalizes curvature and second fundamental form concepts to all closed sets, not just smooth or positive reach sets, and links these to differential properties and support measures.

Findings

Established relation between eigenvalues of approximate differential and principal curvatures.

Provided integral representation for support measures of arbitrary closed sets.

Connected second fundamental form eigenvalues to principal curvatures.

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, \ldots , 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.