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.