Functional perimeter and the dimensional Brunn-Minkowski inequality for log-concave measures

Alexandros Eskenazis, Apostolos Giannopoulos, Natalia Tziotziou

Pith reviewed 2026-05-08 01:50 UTC · model grok-4.3

classification 🧮 math.MG math.FA

keywords log-concave measuresBrunn-Minkowski inequalityisotropic positiongradient estimatesfunctional perimeterconvex setsdimensional inequalities

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The pith

Even log-concave measures satisfy a Brunn-Minkowski inequality for convex sets with exponent at least c over n cubed log n.

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

The paper proves a dimensional version of the Brunn-Minkowski inequality for even log-concave probability measures on Euclidean space. For any symmetric convex sets K and L and any λ between zero and one, the measure of the convex combination raised to the power c_n is at least the convex combination of the individual measures raised to the same power, where c_n is bounded below by a constant over n cubed times the natural log of n. The proof relies on diffusion operators together with a gradient estimate that bounds the integral of the absolute value of the gradient of the log-density by a constant times the dimension. This bound is shown to be optimal in dimension and supplies structural control on the sub-level sets of that gradient, which in turn limits how large the exponent c_n can be made. The same estimate is applied to obtain bounds on weighted perimeters of level sets, projections, moment measures, and surface area measures of isotropic log-concave functions.

Core claim

For every pair of symmetric convex sets K and L in R^n and every λ in (0,1), the inequality μ(λK + (1-λ)L)^{c_n} ≥ λ μ(K)^{c_n} + (1-λ) μ(L)^{c_n} holds, where c_n ≥ c / (n^3 ln n) for an absolute constant c > 0. The proof proceeds by establishing the estimate ∫ |∇ψ| dμ ≤ C n for any isotropic log-concave probability measure μ with density e^{-ψ}, and this estimate is optimal in its dimensional dependence.

What carries the argument

The integral bound ∫ |∇ψ| dμ ≤ C n on the gradient of the logarithmic potential ψ for isotropic even log-concave measures μ, which controls the functional perimeter and supplies the dimensional exponent.

Load-bearing premise

The measures are even and log-concave, and the integral of the absolute gradient of the log-density is at most linear in dimension for the isotropic case.

What would settle it

An explicit even log-concave isotropic measure whose integral of |∇ψ| exceeds any fixed multiple of n would falsify the key estimate and therefore the claimed lower bound on the exponent c_n.

read the original abstract

We study the dimensional Brunn-Minkowski inequality for even log-concave probability measures $\mu$ on $\mathbb{R}^n$ via an analytic approach based on diffusion operators and gradient estimates. Our main result asserts that for every pair of symmetric convex sets $K,L$ in $\mathbb{R}^n$ and every $\lambda\in(0,1)$, $$\mu(\lambda K+(1-\lambda)L)^{c_n} \geq \lambda \mu(K)^{c_n}+(1-\lambda)\mu(L)^{c_n},$$ where $c_n\geq c/n^3\ln n$ for some absolute constant $c>0$. A key ingredient in our proof is the bound $$\int_{\mathbb{R}^n} |\nabla\psi|\,d\mu \leq Cn$$ that we establish for isotropic log-concave probability measures $\mu$ on $\mathbb{R}^n$ with density $e^{-\psi}$, which is optimal in terms of the dimension. This estimate yields structural information on the size of sub-level sets of the gradient of $\psi$ and puts forth a geometric obstruction to further improvements of the Brunn-Minkowski exponent. We also present applications of this estimate to the weighted perimeter of level sets, projections, moment and surface area measures of isotropic log-concave functions, highlighting the central role of the gradient of the logarithmic potential in high-dimensional convexity.

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.

Desk Editor's Note private letter to a colleague

The paper gives an explicit but weak dimensional exponent for the Brunn-Minkowski inequality on even log-concave measures, obtained from a sharp gradient bound proved via diffusion operators.

read the letter

The main contribution is a lower bound c_n ≳ 1/(n^3 ln n) in the dimensional Brunn-Minkowski inequality for even isotropic log-concave measures μ. They reach it by first proving that ∫ |∇ψ| dμ ≤ C n for the potential ψ of an isotropic log-concave density, and this gradient integral is optimal in its dimension dependence. The proof of the bound uses diffusion operators and semigroup gradient estimates, which is a different route from most earlier geometric arguments in the area. From the integral they extract the functional inequality for symmetric convex sets and then list applications to weighted perimeters, projections, moment measures, and surface area measures of isotropic log-concave functions. Those consequences are useful on their own and show the estimate has reach beyond the main inequality. The argument is self-contained: the gradient bound is established independently and then fed into the Brunn-Minkowski step without circularity. Evenness is assumed throughout and matches the symmetric sets K and L, so there is no hidden mismatch. The exponent is the obvious limitation. It decays polynomially and the authors note that their gradient bound prevents any immediate improvement in the n-dependence. Readers who want a constant or even logarithmic exponent will still need other ideas. The work is aimed at people working on quantitative high-dimensional convexity and functional inequalities. The gradient estimate and its listed applications give it concrete value even if the main rate remains modest. It deserves a serious referee.

Referee Report

0 major / 3 minor

Summary. The manuscript claims to prove a dimensional Brunn-Minkowski inequality for even log-concave probability measures μ on R^n via an analytic approach based on diffusion operators and gradient estimates on semigroups. For symmetric convex sets K, L and λ ∈ (0,1), it asserts μ(λK + (1-λ)L)^{c_n} ≥ λ μ(K)^{c_n} + (1-λ) μ(L)^{c_n} where c_n ≥ c/(n^3 ln n) for an absolute c > 0. The key supporting result is the bound ∫ |∇ψ| dμ ≤ C n for isotropic log-concave μ with density e^{-ψ}, claimed to be dimensionally optimal; this is used to obtain structural information on sub-level sets of ∇ψ and to derive applications to weighted perimeters of level sets, projections, moment measures, and surface area measures of isotropic log-concave functions.

Significance. If the central gradient estimate and its application to the inequality hold, the work supplies an explicit (though weak) quantitative dimensional Brunn-Minkowski inequality in the log-concave setting, together with a dimensionally sharp L^1 bound on the gradient of the potential that yields new geometric obstructions and applications. The diffusion-operator method is a constructive strength that may support extensions; the explicit exponent and optimality claim for the gradient integral provide concrete, falsifiable content in asymptotic convex geometry.

minor comments (3)

The dependence of the absolute constant c on the gradient bound should be tracked explicitly from the main inequality statement through the applications section to make the exponent derivation fully transparent.
In the discussion of applications to weighted perimeters and projections, add a short comparison paragraph with prior results on functional perimeters for log-concave measures to clarify the incremental contribution.
Notation: the symbol c_n is introduced in the abstract and main theorem; ensure its precise definition (including the lower bound) appears in a dedicated statement before the proof of the inequality.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive evaluation of our manuscript. The referee's summary accurately reflects our main results on the dimensional Brunn-Minkowski inequality for even log-concave measures and the supporting gradient estimate. We are pleased that the work is viewed as supplying an explicit quantitative inequality together with new geometric applications. Since the report lists no specific major comments, we address the overall assessment below and note that we are happy to incorporate any minor editorial suggestions.

Circularity Check

0 steps flagged

No significant circularity; key bound proven independently

full rationale

The paper derives the dimensional Brunn-Minkowski inequality from the independently established estimate ∫ |∇ψ| dμ ≤ Cn, which is obtained via diffusion operators and gradient estimates on the associated semigroup under the stated isotropic log-concave assumptions. This bound is not defined in terms of the target inequality, nor is it a fitted parameter or self-citation load-bearing step; the evenness and log-concavity hypotheses are applied consistently without reduction to the conclusion. The resulting exponent c_n ≳ 1/(n^3 ln n) follows directly as a consequence of the dimension dependence in the proven bound, with no self-definitional or renaming circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard domain assumptions of convex geometry and analysis; no free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Even log-concave probability measures have symmetric convex potential ψ such that the density is e^{-ψ}.
Invoked throughout the statement of the main result and the gradient estimate.
domain assumption Isotropic position normalizes the measure so that the gradient bound can be stated with dimension-linear constant.
Required for the key estimate ∫ |∇ψ| dμ ≤ C n.

pith-pipeline@v0.9.0 · 5562 in / 1494 out tokens · 54918 ms · 2026-05-08T01:50:00.554495+00:00 · methodology

discussion (0)

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