Maximum and minimum of support functions

Abstract

For given continuous functions $\gamma_{{}_{i}}: S^{n}\to \mathbb{R}_{+}$ (where $i=1, 2$), the functions $\gamma_{{}_{max}}$ and $\gamma_{{}_{min}}$ can be defined as natural way. In this paper, we show that the Wulff shape associated to $\gamma_{{}_{max}}$ is the convex hull of the union of Wulff shapes associated to $\gamma_{{}_1}$ and $\gamma_{{}_2}$ , if $\gamma_{{}_1}$ and $\gamma_{{}_2}$ are convex integrands. And, the Wulff shape associated to $\gamma_{{}_{min}}$ is the intersection of Wulff shapes associated to $\gamma_{{}_1}$ and $\gamma_{{}_2}$. Moreover, relationships between their dual Wulff shapes are given.