Topological aspect of Wulff shapes

Abstract

In this paper we investigate Wulff shapes in $\mathbb{R}^{n+1}$ $(n\ge 0)$ from the topological viewpoint. A topological characterization of the limit of Wulff shapes and the dual Wulff shape of the given Wulff shape are provided. Moreover, we show that the given Wulff shape is never a polytope if its support function is of class $C^1$. Several characterizations of the given Wulff shape from the viewpoint of pedals are also provided. One of such characterizations may be regarded as a bridge between the mathematical aspect of crystals at equilibrium and the mathematical aspect of perspective projections.