Behavior of convex integrand at apex of its Wulff shape

Abstract

Let $\gamma: S^n\to \mathbb{R}_+$ be a convex integrand and $\mathcal{W}_\gamma$ be the Wulff shape of $ \gamma$. Apex point naturally arise in non-smooth Wulff shape, in particular, vertex of convex polytope. %Let $P\in S^n$. In this paper, we study the behavior of convex integrand around apex point of its Wulff shape. We prove that $\gamma(P)$ is locally maximum, and $\mathbb{R}_+ P\cap \partial \mathcal{W}_\gamma$ is an apex point of $\mathcal{W}_\gamma$ if and only if the graph of $\gamma$ around the apex point is a pice of sphere. 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 bides of constant width $\tau$, whose boundary consists only of arcs of circles of radius $\tau-\frac{\pi}{2}$ and great circle segments such that $\lim_{I\to \infty}C_i=C$ with respect to the Hausdorff distance.