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arXiv · 2607.16924

The Spherical Gr\"unbaum Inequality

Sergii Myroshnychenko·Dmitry Ryabogin·Kateryna Tatarko·Vladyslav Yaskin
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Abstract

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 $$\sigma(K\cap u^+) \ge \left(1-\frac{1}{n}\right)^{n-1} \sigma(K),$$ where $\sigma$ denotes the spherical measure. The constant in this inequality is optimal.

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Sergii Myroshnychenko, Dmitry Ryabogin, Kateryna Tatarko, Vladyslav Yaskin. 2026-07-18. The Spherical Gr\"unbaum Inequality. https://arxiv.org/abs/2607.16924

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