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.

arXiv comment13 pages, 6 figures