Fine properties of the curvature of arbitrary closed sets

Given an arbitrary closed set A of Rn, 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 Zähle. Then, we provide for every m= 1 , … , n- 1 an integral representation for the support measure μm of A with respect to the m-dimensional Hausdorff measure. Moreover, a notion of second fundamental form QA for an arbitrary closed set A is introduced so that the finite principal curvatures of A correspond to the eigenvalues of QA. Finally, we establish the relation between QA and the approximate differential of order 2 for sets introduced in a previous work of the author, proving that in a certain sense the latter corresponds to the absolutely continuous part of QA.