## A. Chambolle - L. Lussardi - E. Villa

# Anisotropic tubular neighborhoods of sets

created by lussardi on 24 Jul 2019

modified on 13 Nov 2020

[

BibTeX]

*Accepted Paper*

**Inserted:** 24 jul 2019

**Last Updated:** 13 nov 2020

**Journal:** Math. Z.

**Pages:** 1-19

**Year:** 2020

**Abstract:**

Let $E \subset \mathbb R^N$ be a compact set and $C\subset \mathbb R^N$ be a convex body with $0\in{\rm int}\,C$. We prove that the topological boundary of the anisotropic enlargement $E+rC$ is contained in a finite union of Lipschitz surfaces. We also investigate the regularity of the volume function $V_E(r):=

E+rC

$ proving a formula for the right and the left derivatives at any $r>0$ which implies that $V_E$ is of class $C^1$ up to a countable set completely characterized. Moreover, some properties on the second derivative of $V_E$ are proved.

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