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

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--Santaló 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.