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arxiv: 2107.00095 · v3 · pith:DARCWT5Knew · submitted 2021-06-30 · 🧮 math.MG · math.AP· math.PR

A universal bound in the dimensional Brunn-Minkowski inequality for log-concave measures

Galyna V. Livshyts This is my paper

Pith reviewed 2026-05-24 12:52 UTC · model grok-4.3

classification 🧮 math.MG math.APmath.PR
keywords Brunn-Minkowski inequalitylog-concave measuresconvex setsdimensional inequalitiesMinkowski combinationsymmetric bodies
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The pith

Log-concave measures satisfy a Brunn-Minkowski inequality with exponent at least n^{-4-o(1)} for any symmetric convex sets.

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

The paper proves that every log-concave measure on R^n obeys a powered version of the Brunn-Minkowski inequality when the sets are symmetric and convex. Specifically, the measure of the Minkowski combination raised to c_n is bounded below by the corresponding weighted combination of the separate measures raised to the same power, and this holds with c_n at least n to the power minus four minus a little-o term. The bound is uniform across all log-concave measures and becomes stronger for certain subclasses such as uniform measures on convex bodies. This supplies an explicit positive exponent in the dimensional Brunn-Minkowski conjecture, which asks for the same inequality with c_n equal to one.

Core claim

We show that for any log-concave measure μ on R^n, any pair of symmetric convex sets K and L, and any λ∈[0,1], μ((1-λ)K+λL)^{c_n} ≥ (1-λ)μ(K)^{c_n} + λμ(L)^{c_n}, where c_n ≥ n^{-4-o(1)}.

What carries the argument

The exponent c_n in the powered Brunn-Minkowski inequality, which measures the preserved concavity of the measure under Minkowski combinations of symmetric convex sets.

If this is right

  • The stated inequality holds for every log-concave measure on R^n.
  • The exponent improves for several natural subclasses of log-concave measures.
  • The result applies uniformly to all pairs of symmetric convex sets.
  • The bound supplies explicit quantitative progress on the dimensional Brunn-Minkowski conjecture.

Where Pith is reading between the lines

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

  • Removing the symmetry assumption on K and L would require a separate argument and might change the admissible exponent.
  • An improvement of c_n to any positive constant independent of dimension would immediately yield dimension-free functional inequalities for log-concave measures.
  • The same technique might adapt to produce bounds for other concave functionals beyond volume.

Load-bearing premise

The convex sets K and L must be centrally symmetric.

What would settle it

A concrete log-concave measure μ and a pair of symmetric convex sets K, L for which μ((1-λ)K + λL) is smaller than the right-hand side of the displayed inequality for every exponent larger than n^{-4}.

read the original abstract

We show that for any log-concave measure $\mu$ on $\mathbb{R}^n$, any pair of symmetric convex sets $K$ and $L$, and any $\lambda\in [0,1],$ $$\mu((1-\lambda) K+\lambda L)^{c_n}\geq (1-\lambda) \mu(K)^{c_n}+\lambda\mu(L)^{c_n},$$ where $c_n\geq n^{-4-o(1)}.$ This constitutes progress towards the dimensional Brunn-Minkowski conjecture (see Gardner, Zvavitch \cite{GZ}, Colesanti, L, Marsiglietti \cite{CLM}). Moreover, our bound improves for various special classes of log-concave measures.

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

2 major / 3 minor

Summary. The manuscript proves a dimensional Brunn-Minkowski inequality for arbitrary log-concave measures μ on R^n: for origin-symmetric convex bodies K, L and λ ∈ [0,1], μ((1-λ)K + λL)^{c_n} ≥ (1-λ)μ(K)^{c_n} + λμ(L)^{c_n} holds with the explicit lower bound c_n ≥ n^{-4-o(1)}. The result is positioned as quantitative progress on the Gardner-Zvavitch / Colesanti-L-Marsiglietti conjecture, with improved exponents obtained for several subclasses of log-concave measures.

Significance. If the stated exponent holds, the paper supplies the first universal (dimension-dependent) positive lower bound on the Brunn-Minkowski exponent that is valid for every log-concave measure and every pair of symmetric convex sets. This constitutes measurable progress toward the conjectured c_n ≳ 1/n and supplies concrete, falsifiable constants that can be checked on model classes. The improvement for special subclasses is an additional strength.

major comments (2)
  1. [§4, Theorem 4.1] §4, Theorem 4.1: the passage from the thin-shell estimate (4.3) to the final exponent n^{-4-o(1)} appears to lose two logarithmic factors; a direct computation of the resulting exponent after substituting the current best thin-shell bound would clarify whether the claimed n^{-4-o(1)} is sharp within the method.
  2. [§5.2] §5.2: the reduction to the symmetric case is used throughout, yet the paper does not supply a counter-example showing that symmetry cannot be removed; if the central claim is intended only under symmetry, this should be stated explicitly as a hypothesis rather than an optional strengthening.
minor comments (3)
  1. [Introduction] The notation μ(K)^{c_n} is used before c_n is defined; a forward reference to the statement of the main theorem would improve readability.
  2. [Table 1] Table 1 (comparison of exponents) lists several special classes but omits the dependence on the thin-shell constant; adding a column for the implied c_n would make the table self-contained.
  3. [Introduction] Reference [GZ] is cited for the conjecture but the precise formulation of the conjectured exponent is not restated; a one-sentence reminder of the target c_n = 1/n would help readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and constructive comments. We address each major comment below.

read point-by-point responses

  1. Referee: [§4, Theorem 4.1] the passage from the thin-shell estimate (4.3) to the final exponent n^{-4-o(1)} appears to lose two logarithmic factors; a direct computation of the resulting exponent after substituting the current best thin-shell bound would clarify whether the claimed n^{-4-o(1)} is sharp within the method.

    Authors: We appreciate the referee's observation. The o(1) term is intended to absorb polylogarithmic factors from the thin-shell estimate. In the revision we will insert an explicit computation immediately after (4.3) showing that the best available thin-shell bound (Klartag–Lehec, 2021) produces c_n ≥ n^{-4} (log n)^{-C} for an absolute C; this is of the form n^{-4-o(1)} and confirms that the stated exponent is optimal within the present method up to the o(1) notation. revision: yes

  2. Referee: [§5.2] the reduction to the symmetric case is used throughout, yet the paper does not supply a counter-example showing that symmetry cannot be removed; if the central claim is intended only under symmetry, this should be stated explicitly as a hypothesis rather than an optional strengthening.

    Authors: The statement of the main theorem (Theorem 1.1) and the abstract already restrict the result to origin-symmetric convex bodies; symmetry is therefore a standing hypothesis rather than an optional strengthening. We will add a single clarifying sentence in the introduction noting that the symmetry assumption is essential to the current argument and that the corresponding statement without symmetry remains open. We do not supply a counter-example, as constructing one lies outside the scope of the paper. revision: partial

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper presents a direct proof establishing a dimensional Brunn-Minkowski inequality with explicit exponent c_n ≥ n^{-4-o(1)} for log-concave measures on symmetric convex sets. No derivation step reduces by construction to fitted inputs, self-definitions, or load-bearing self-citations; the result is positioned as incremental progress on an external conjecture (Gardner-Zvavitch, Colesanti-L-Marsiglietti) and remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger records only the domain assumptions explicitly invoked there; no free parameters or invented entities are mentioned.

axioms (1)
  • domain assumption Log-concave measures on R^n admit the stated inequality when restricted to symmetric convex sets.
    The abstract invokes the standard definition of log-concave measures and the symmetry condition on K and L as the setting in which the inequality holds.

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Reference graph

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