The spherical dual transform is an isometry for spherical Wulff shapes

TL;DR

This paper proves that the spherical dual transform preserves distances between spherical Wulff shapes under the Pompeiu-Hausdorff metric, establishing it as an isometry.

Contribution

It demonstrates that the spherical dual transform is an isometry for spherical Wulff shapes, a novel geometric property in this context.

Findings

The spherical dual transform is an isometry under the Pompeiu-Hausdorff metric.

Spherical Wulff shapes have unique duals with preserved distances.

The result extends understanding of geometric transformations in spherical shape theory.

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.