# Maximum and minimum of support functions

## TL;DR

This paper investigates how maximum and minimum support functions relate to Wulff shapes, showing that the shape for the maximum is the convex hull of the union, and for the minimum is the intersection, with dual relationships explored.

## Contribution

It establishes geometric relationships between support functions and Wulff shapes for maximum and minimum functions, extending understanding of convex integrands.

## Key findings

- Wulff shape for max support function is convex hull of union.
- Wulff shape for min support function is intersection of shapes.
- Dual relationships between Wulff shapes are characterized.

## 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.