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.
Categories: math.MG