For any Wulff shape, its dual Wulff shape is naturally defined. A self-dual Wulff shape is a Wulff shape equaling its dual Wulff shape exactly. In this paper, it is shown that a Wulff shape is self-dual if and only if the spherical convex body induced by it is of constant width π/2. 2010 Mathematics Subject Classification. 52A55.

Abstract. In this paper, it is shown that a Wulff shape is strictly convex if and only if its convex integrand is of class C 1 . Moreover, applications of this result are given.

This is a survey article on the spherical method for studying Wulff shapes and related topics. The spherical method, which seems less common, is a powerful tool to study Wulff shapes and their related topics. It is verified how powerful the spherical method is by various results which seem difficult to be obtained without using the method. In this survey, the spherical method is explained in detail, and results obtained by using the spherical method until April 2016 are explained as well.A. Dinghas (1944, [14]) gave a formal proof. J. Taylor (1978, [69]) gave a precise proof for very general surface energies and a very general class of set for which the surface energy is defined by using geometric measure theory. B. Dacorogna and C. E. Pfister (1992, [13]) gave an analytic proof when n = 1. I. Fonseca (1991, [16]) and I. Fonseca and S. Müller (1991, [17]) gave a simpler proof for arbitrary dimensions.Let Γ γ be the boundary of the convex hull of inv(graph(γ)). Then, the following is known.Proposition 1 ([69, 23]). Let γ 1 , γ 2 be two elements of C 0 (S n , R + ) such that Γ γ1 = Γ γ2 . Then, the equality W γ1 = W γ2 holds.By Proposition 1, counterexamples for Question 3 are easily constructed (see Figure 3).As the next question, the following question naturally arises.

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.

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