Anisotropic tubular neighborhoods of sets

Let E⊂ RN be a compact set and C⊂ RN be a convex body with 0∈intC. 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 VE(r) : = | E+ rC| proving a formula for the right and the left derivatives at any r> 0 which implies that VE is of class C1 up to a countable set completely characterized. Moreover, some properties on the second derivative of VE are proved.