Convex geometric reasoning for crystalline energies

TL;DR

This paper offers a simplified, convex geometric approach to the classical Wulff problem for crystalline energies, providing insights into the shape of crystals with finitely faceted structures.

Contribution

It introduces a direct Minkowski Theory-based method to analyze the Wulff problem for crystalline integrands, simplifying previous geometric measure theory proofs.

Findings

Provides a new convex geometric proof of the Wulff theorem for crystalline energies.

Clarifies the shape characterization of crystals with finitely faceted structures.

Enhances understanding of crystalline surface energies through Minkowski Theory.

Abstract

The present work revisits the classical Wulff problem restricted to crystalline integrands, a class of surface energies that gives rise to finitely faceted crystals. The general proof of the Wulff theorem was given by J.E. Taylor (1978) by methods of Geometric Measure Theory. This work follows a simpler and direct way through Minkowski Theory by taking advantage of the convex properties of the considered Wulff shapes.