The Wulff construction for convex integrands

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.