The spherical dual transform is an isometry for spherical Wulff shapes

Abstract

A spherical Wulff shape is the spherical counterpart of a Wulff shape which is the well-known geometric model of a crystal at equilibrium introduced by G. Wulff in 1901. As same as a Wulff shape, each spherical Wulff shape has its unique dual. The spherical dual transform for spherical Wulff shapes is the mapping which maps a spherical Wulff shape to its spherical dual Wulff shape. In this paper, it is shown that the spherical dual transform for spherical Wulff shapes is an isometry with respect to the Pompeiu-Hausdorff metric.