Anisotropic tubular neighborhoods of sets

Let $$E \subset {{\mathbb {R}}}^N$$ be a compact set and $$C\subset {{\mathbb {R}}}^N$$ be a convex body with $$0\in \mathrm{int}\,C$$ . 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.