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.