The Wulff construction for convex integrands

TL;DR

This paper establishes a precise relationship between the convex integrand functions of Wulff shapes and their geometric distances, providing a new way to compare shapes via functional analysis.

Contribution

It proves that the maximum distance between convex integrands equals the Pompeiu-Hausdorff distance between Wulff shapes, linking shape geometry with function space metrics.

Findings

The equality $d(\gamma_{W_1}, \gamma_{W_2})= h(W_1, W_2)$ is established.

The relationship provides a new method to compare Wulff shapes.

Applications of the main result are demonstrated.

Abstract

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.