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, \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.