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arxiv: 2606.30579 · v1 · pith:KYZBNX6Knew · submitted 2026-06-29 · 🧮 math.AP

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

Shibing Chen , Yuanyuan Li , Dongmeng Xi , Zhe-Feng Xu This is my paper

Pith reviewed 2026-06-30 04:41 UTC · model grok-4.3

classification 🧮 math.AP
keywords Blaschke-Santaló inequalitymany-body inequalitymulti-marginal optimal transportorigin-symmetric setsvolume product boundpseudo-Euclidean estimatefunctional inequalities
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The pith

Origin-symmetric measurable sets in R^n whose pairwise inner products sum to at most binom(k,2) satisfy the sharp volume-product bound ∏|K_i| ≤ |B^n|^k.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves a many-body extension of the Blaschke-Santaló inequality that bounds the product of the volumes of k origin-symmetric sets by the k-th power of the volume of the Euclidean unit ball, provided the sets obey a uniform upper bound on the sum of all their pairwise inner products. The result is shown to be sharp and all equality cases are characterized. The argument proceeds by establishing the geometric form first via multi-marginal optimal transport and a pseudo-Euclidean volume comparison, after which an existing geometric-functional equivalence immediately yields the corresponding functional inequality proposed by Kolesnikov and Werner. A reader interested in convex geometry or optimal transport would care because the bound supplies a concrete, dimension-independent volume control under an explicit linear constraint on the sets.

Core claim

Let K_1,…,K_k ⊂ R^n be origin-symmetric measurable sets of finite volume such that ∑_{1≤i<j≤k} ⟨x_i,x_j⟩ ≤ binom(k,2) for all x_i ∈ K_i and x_j ∈ K_j. Then ∏_{i=1}^k |K_i| ≤ |B^n|^k, with equality cases fully characterized. The proof combines multi-marginal optimal transport with a pseudo-Euclidean volume estimate; the geometric-functional equivalence of Kalantzopoulos and Saroglou then transfers the result to the functional setting.

What carries the argument

Multi-marginal optimal transport together with a pseudo-Euclidean volume estimate, which produces the geometric inequality from which the functional version follows via the cited equivalence.

If this is right

  • The functional many-body Blaschke-Santaló inequality proposed by Kolesnikov and Werner holds.
  • Equality in the geometric inequality is attained precisely in the cases characterized by the argument.
  • The bound is dimension-independent and sharp for every n and k.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same transport-plus-volume-estimate strategy may extend to other linear constraints on multiple convex bodies.
  • Equality cases likely consist of suitably scaled and rotated copies of the unit ball; verifying this characterization explicitly would confirm uniqueness.
  • The result supplies a new test case for numerical optimal-transport solvers in high dimensions.

Load-bearing premise

The geometric-functional equivalence of Kalantzopoulos and Saroglou holds so that the functional inequality follows from the geometric one.

What would settle it

An explicit collection of origin-symmetric sets K_1 through K_k satisfying the pairwise inner-product sum condition yet having ∏|K_i| > |B^n|^k would disprove the claimed bound.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript establishes the sharp many-body Blaschke-Santaló inequality ∏_{i=1}^k |K_i| ≤ |B^n|^k for origin-symmetric measurable sets K_i ⊂ ℝ^n of finite volume satisfying the pairwise inner-product constraint ∑_{1≤i<j≤k} ⟨x_i, x_j⟩ ≤ binom(k,2) for all x_i ∈ K_i, x_j ∈ K_j. The proof combines multi-marginal optimal transport with a pseudo-Euclidean volume estimate; equality cases are characterized. Via the geometric-functional equivalence of Kalantzopoulos-Saroglou, the corresponding functional inequality of Kolesnikov-Werner is also obtained.

Significance. If the arguments hold, the result resolves the many-body conjecture of Kalantzopoulos-Saroglou and supplies a new application of multi-marginal optimal transport to convex geometry. The equality characterization and the passage to the functional setting are additional strengths. The approach is direct and avoids parameter fitting.

minor comments (3)
  1. The abstract states that the proof 'combines multi-marginal optimal transport with a pseudo-Euclidean volume estimate,' but the manuscript should include a short roadmap paragraph (e.g., after the statement of Theorem 1.1) that explicitly identifies which transport plan is constructed and where the volume estimate is applied.
  2. Notation for the pseudo-Euclidean volume (presumably introduced in §3 or §4) should be defined at first use and cross-referenced when the estimate is invoked in the main argument.
  3. The equality-case analysis would benefit from a separate subsection that isolates the conditions under which the multi-marginal plan becomes a product measure and the volume estimate saturates.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our manuscript, the recognition of its significance in resolving the many-body conjecture, and the recommendation for minor revision. No major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation uses external tools and non-author citation

full rationale

The paper establishes the geometric inequality directly via multi-marginal optimal transport combined with a new pseudo-Euclidean volume estimate, then invokes the geometric-functional equivalence from Kalantzopoulos and Saroglou (distinct authors) to obtain the functional version. No load-bearing step reduces by construction to a fitted input, self-definition, or self-citation chain; the central result is obtained from independent external methods whose assumptions do not presuppose the target inequality. This is the normal case of a self-contained derivation against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The claim rests on standard properties of Lebesgue measure, existence of multi-marginal optimal transport plans, and the cited geometric-functional equivalence; the pseudo-Euclidean volume estimate is the novel technical step whose details are absent from the abstract.

axioms (3)
  • standard math Lebesgue measure on R^n is translation- and rotation-invariant
    Required to define and compare volumes |K_i| and |B^n|
  • domain assumption Multi-marginal optimal transport problems admit solutions for the given measurable sets
    Central tool invoked in the proof method
  • domain assumption The geometric-functional equivalence of Kalantzopoulos and Saroglou holds
    Used to obtain the functional version from the geometric inequality

pith-pipeline@v0.9.1-grok · 5685 in / 1519 out tokens · 75228 ms · 2026-06-30T04:41:13.782003+00:00 · methodology

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