The Spherical Grünbaum Inequality

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 $\theta \in {\mathbb{S}}^{n-1}$. Then for any $u\in {\mathbb{S}}^{n-1}$ that is orthogonal to $\theta$ we have $$\sigma (K\cap u^{+})\ge {(1-\frac{1}{n})}^{n-1}\sigma (K),$$ where $\sigma$ denotes the spherical measure. The constant in this inequality is optimal.