Uniqueness of the surface energy density for a Wulff shape with $C^1$ boundary

TL;DR

This paper proves that if a Wulff shape has a $C^1$ boundary, then its defining surface energy density function is uniquely determined as the convex integrand, establishing a key geometric property.

Contribution

It demonstrates the uniqueness of the surface energy density function for Wulff shapes with $C^1$ boundaries, linking boundary smoothness to the convex integrand.

Findings

If the Wulff shape boundary is $C^1$, then the surface energy density is the convex integrand.

The result establishes a geometric characterization of the surface energy density.

The paper provides a condition for the uniqueness of the defining function of Wulff shapes.

Abstract

Let $\gamma: S^n\to \mathbb{R}_+$ be a continuous function and let $\mathcal{W}_\gamma$ be the Wulff shape associated with $\gamma$. In this paper, we show that if the boundary of $\mathcal{W}_\gamma$ is a $C^1$ submanifold, then $\gamma$ must be the convex integrand of $\mathcal{W}_\gamma$.