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

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