S. Myroshnychenko

The Spherical Gr\"unbaum Inequality

We prove an analogue of Gr\"unbaum's inequality on the sphere. Let $n \geq 3$ 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 \left(1-\frac{1}{n}\right)^{...