Strictly convex Wulff shapes and 𝐶¹ convex integrands

Abstract

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.

Limit of the Hausdorff distance for one-parameter families of Wulff shapes constructed by affine perturbations of dual Wulff shapes

“…Proposition 2. For any i ∈ {1, 2}, suppose that 3 2 ||a i || < c i . Let g ai,ci : S n → R + be defined by g ai,ci (X) = f ai,ci (X).…”

Self-dual Wulff shapes and spherical convex bodies of constant width ${\pi}/{2}$

“…On the other hand, any Reuleaux triangle in R 2 containing the origin as an interior point (see Figure 3) is not a self-dual Wulff shape, although it is a Wulff shape of constant width in R 2 . This is because any Reuleaux triangle is strictly convex, and thus the boundary of it must be smooth by [1] if it is self-dual. However, there are three non-smooth points for any Reuleaux triangle in R 2 .…”

Spherical method for studying Wulff shapes and related topics

“…In the case k = 1, a partial answer to Question 8 can be found in [51]. In [23], a complete answer to this question with a rigorous proof is obtained, and it is summarized in Subsection 3.4.…”

“…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]).…”

“…Proposition 2. [69,23] Let W be an element of H conv,0 ( R n+1 ) . Then, for any γ ∈ W −1 (W ) and any θ ∈ S n , the following inequality holds.…”