Maximum and minimum of support functions

Abstract

For given continuous functions $\gamma_{{}_{i}}: S^{n}\to \mathbb{R}_{+}$ ($i=1, 2$), the functions $\gamma_{{}_{\max}}$ and $\gamma_{{}_{\min}}$ can be defined naturally. In this paper, by applying the spherical method, we first 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. Next, we show that 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.