*Published Paper*

**Inserted:** 22 dec 2001

**Journal:** Arch. Rational Mech. Anal.

**Volume:** 157

**Number:** 3

**Pages:** 193-217

**Year:** 2001

**Abstract:**

For a nonsmooth positively one homogeneous convex function $\phi : \ensuremath{\mathbb R^n} \to [0,+\infty[$, it is possible to introduce the class ${\mathcal R}_\phi(\ensuremath{\mathbb R^n})$ of smooth boundaries with respect to $\phi$, to define their $\phi$-mean curvature $\kappa_\phi$, and to prove that, for $E \in {\mathcal R}_\phi(\ensuremath{\mathbb R^n})$, there holds $\kappa_\phi \in L^\infty(\dE)$ \cite{BeNoPa1:00}. Based on these results, we continue the analysis on the structure of $\partial E$ and on the regularity properties of $\kappa_\phi$. We prove that a facet $F$ of $\partial E$ is Lipschitz (up to negligible sets) and that $\kappa_\phi$ has bounded variation on $F$. Further properties of the jump set of $\kappa_\phi$ are inspected: in particular, in three space dimensions, we relate the sublevel sets of $\kappa_\phi$ on $F$ with the geometry of the Wulff shape $\mathcal{W}_\phi}:= \{\p \leq 1\}$.