The Wulff construction for convex integrands

Abstract

For any given Wulff shape $\mathcal{W}$, we can define the unique continuous function $S^n\to {\mathbb{R}}_{+}$ called convex integrand, denoted by ${\gamma }_{{}_{\mathcal{W}}}$. In this paper, we show that, for any Wulff shapes ${\mathcal{W}}_1$ and ${\mathcal{W}}_2$, the equality $d({\gamma }_{{}_{{\mathcal{W}}_1}},{\gamma }_{{}_{{\mathcal{W}}_2}})=h({\mathcal{W}}_1,{\mathcal{W}}_2)$ holds, where $d$ is the maximum distance of the function space consisting of convex integrands and $h$ is the Pompeiu-Hausdorff distance of the space consisting of Wulff shapes. Moreover, applications of this result are given.

6 pages, 2 figures. This paper has been withdrawn by the author due to important improvements to be done