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.2010 Mathematics Subject Classification. 47N10, 52A30, 82D25.

Spherical Separation Theorem

“…Our motivation for proving Theorem 1 is coming from fruitful discussions with the reviewer of Mathematical Reviews for the authors' paper [2]. As shown in the review [5], some arguments of [2] are not clear.…”

Spherical method for studying Wulff shapes and related topics

“…It is easily seen that Proposition 10 holds for any n ∈ N. In the next subsection, a result obtained in [22] which is stronger than Proposition 10 (for general n) is surveyed. Namely, in Subsection 3.3, it shall be stated that the spherical polar transform ⃝ :…”

“…The mapping Ψ N has been already used to study many topics related to perpendicularity. For instance, it was used for studying singularities of spherical pedal curves in [36,53,54,55], for studying spherical pedal unfoldings in [56], for studying hedgehogs and no-silhouettes in [59], for studying (spherical) Wulff shapes in [60,22,23], and for studying the aperture of plane curves in [35]. A hyperbolic version of Ψ N is also useful in hyperbolic situation (see [34]).…”

“…One of essential tools in Subsection 3.2 was treating any spherical polar set as the image of a non-empty closed subset by the map called the spherical polar transform. In [22], the spherical polar transform is investigated more and more by using the spherical method explained in Section 2. In this subsection, we survey results on the spherical polar transform obtained in [22].…”

“…In [22], the spherical polar transform is investigated more and more by using the spherical method explained in Section 2. In this subsection, we survey results on the spherical polar transform obtained in [22].…”