Behavior of convex integrand at a d-apex of its Wulff shape and approximation of spherical bodies of constant width

Abstract

Let \(\gamma : S^n\rightarrow \mathbb {R}_+\) be a convex integrand and \(\mathcal {W}_\gamma \) be the Wulff shape of \(\gamma \). A d-apex point naturally arises in a non-smooth Wulff shape, in particular, as a vertex of a convex polytope. In this paper, we study the behavior of the convex integrand at a d-apex point of its Wulff shape. We prove that \(\gamma (P)\) is locally maximum, and \(\mathbb {R}_+ P\cap \partial \mathcal {W}_\gamma \) is a d-apex point of \(\mathcal {W}_\gamma \) if and only if the graph of \(\gamma \) around the d-apex point is a piece of a sphere with center \(\frac{1}{2}\gamma (P)P\) and radius \(\frac{1}{2}\gamma (P)\). As an application of the proof of this result, we prove that for any spherical convex body C of constant width \(\tau >\pi /2\), there exists a sequence \(\{C_i\}_{i=1}^\infty \) of convex bodies of constant width \(\tau \), whose boundaries consist only of arcs of circles of radius \(\tau -\frac{\pi }{2}\) and great circle arcs such that \(\lim _{i\rightarrow \infty }C_i=C\) with respect to the Hausdorff distance.