*Published Paper*

**Inserted:** 21 jul 2020

**Last Updated:** 23 jul 2020

**Journal:** Ann. Mat. Pura Appl.

**Volume:** 199(2020)

**Year:** 2017

**Doi:** 10.1007/s10231-019-00926-w

**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 Zähle. Then, we provide for every $ m=1,…,n−1 $ an integral representation for the support measure $ \mu_m$ of A with respect to the m-dimensional Hausdorff 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 $Q_A$. Finally, we establish the relation between $Q_A$ 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 $Q_A$.