The Spherical Grünbaum Inequality

Abstract

We prove an analogue of Grünbaum's inequality on the sphere. Let $n\ge 3$ and let $K$ be a convex body on ${\mathbb{S}}^{n-1}\subset {\mathbb{R}}^n$ with centroid at $\hat{I}¸\in {\mathbb{S}}^{n-1}$. Then for any $u\in {\mathbb{S}}^{n-1}$ that is orthogonal to $\hat{I}¸$ we have $$\ddot{I}ƒ(K\cap u^{+})\ge {\left(1-\frac{1}{n}\right)}^{n-1}\ddot{I}ƒ(K),$$ where $\ddot{I}ƒ$ denotes the spherical measure. The constant in this inequality is optimal.

19 pages, 5 figures