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arXiv · 2606.30579

The many-body Blaschke-Santal\'o type inequality via optimal transport

Shibing Chen·Yuanyuan Li·Dongmeng Xi·Zhe-Feng Xu
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Abstract

Let $K_1,\ldots,K_k\subset\mathbb R^n$ be origin-symmetric measurable sets of finite volume such that \[ \sum_{1\le i<j\le k}\langle x_i,x_j\rangle\le \binom{k}{2}, \qquad \forall\,x_i\in K_i, x_j\in K_j. \] We prove the sharp many-body Blaschke--Santal\'o type inequality \[ \prod_{i=1}^k |K_i|\le |B^n|^k \] proposed by Kalantzopoulos and Saroglou, and characterize all equality cases. The proof combines multi-marginal optimal transport with a pseudo-Euclidean volume estimate. Using the geometric--functional equivalence of Kalantzopoulos and Saroglou, we also establish the functional version inequality proposed by Kolesnikov and Werner.

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Shibing Chen, Yuanyuan Li, Dongmeng Xi, Zhe-Feng Xu. 2026-06-29. The many-body Blaschke-Santal\'o type inequality via optimal transport. https://arxiv.org/abs/2606.30579

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