The Spherical Grünbaum Inequality

2026/07/18 by Sergii Myroshnychenko, Dmitry Ryabogin, Kateryna Tatarko +1
#math.MG #math.DG #math.FA

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Abstract

We prove an analogue of Grünbaum's inequality on the sphere. Let n ≥ 3 and let K be a convex body on \mathbb S^n-1⊂ \mathbb Rⁿ with centroid at θ∈ \mathbb S^n-1. Then for any u∈ \mathbb S^n-1 that is orthogonal to θ we have σ(K∩ u⁺) ≥ (1-(1)/(n))^n-1 σ(K), where σ denotes the spherical measure. The constant in this inequality is optimal.

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