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

Abstract: For every $n\ge 2$, we prove that there exists an exponent $p_n$ such that, for every even log-concave probability measure $μ$ on $\mathbb R<sup>n$, all nonempty symmetric convex sets $K,L\subseteq\mathbb R<sup>n$, and all $λ\in[0,1]$, $$μ(λK+(1-λ)L)^{p_n} \ge λμ(K)^{{p_n}+(1-λ)μ(L)^{p_n},}$$ where $p_n\ge \frac{c}{n^2\ln n}$ for some absolute constant $c&gt;0$.

• The paper establishes a new uniform lower bound of 1/(n² ln n) for the exponent in the dimensional Brunn–Minkowski inequality for even log-concave measures.
• It uses analytic and variational techniques, including energy-type estimates and regularization, to overcome obstacles in convexity in high-dimensional settings.
• The results improve previous bounds, bridging functional and geometric inequalities in convex geometry with implications for information theory and high-dimensional analysis.

A Uniform Bound in the Dimensional Brunn–Minkowski Inequality for Even Log-Concave Measures

Introduction and Context

The paper "A uniform bound in the dimensional Brunn–Minkowski inequality for even log-concave measures" (2607.10104) addresses a central problem in high-dimensional convex geometry: the generalization of the Brunn–Minkowski inequality to the setting of general, even log-concave measures. The Brunn–Minkowski inequality is a foundational result linking the geometry of convex bodies and the measure of their Minkowski sums. For Lebesgue measure, it asserts $n$-concavity with respect to Minkowski interpolation. Extending such dimensional concavity to more general classes of measures, and notably to all even log-concave measures, forms a major open conjecture with deep repercussions in convex geometry, analysis, and information theory.

The dimensional Brunn–Minkowski conjecture posits a universal $1/n$-concavity of measure for symmetric convex sets and even log-concave measures. While affirmative for Gaussian, rotationally invariant, and certain unconditional measures, a general resolution remains elusive for $n \geq 3$. This paper advances the state-of-the-art, improving prior uniform lower bounds for the exponent in the power-concavity inequality for such measures.

Main Results and Technical Contributions

The principal theorem establishes a new lower bound for the exponent in the dimensional Brunn–Minkowski inequality for every dimension $n \geq 2$. Specifically, for every even log-concave probability measure $\mu$ on $\mathbb{R}^n$, and all non-empty symmetric convex sets $K, L \subseteq \mathbb{R}^n$, the inequality

$$\mu(\lambda K + (1-\lambda) L)^{p_n} \geq \lambda \mu(K)^{p_n} + (1-\lambda) \mu(L)^{p_n}$$

holds for all $\lambda \in [0,1]$ with a uniform exponent

$p_n \geq \frac{c}{n^2 \ln n}$

where $1/n$0 is an absolute constant.

This result improves the previously best-known lower bound of $1/n$1 (Eskenazis, Giannopoulos, Tziotziou 2026 (Eskenazis et al., 4 May 2026)) by a factor of $1/n$2. The prior universal bound $1/n$3 was shown in Livshyts (2023). The paper achieves this improvement by developing new energy-type estimates that sidestep known obstacles in the analytic method related to the loss of convexity for low-gradient level sets.

Methodology

The authors employ analytic and variational techniques for log-concave measures, structured around several elements:

1. Reduction to Isotropic Case: Via affine transformation and measure push-forward, every log-concave measure is reduced to its isotropic representative, where analytic inequalities are best controlled.
2. Poincaré and Logarithmic Potential Estimates: The analysis leverages sharp bounds for the Poincaré constant of isotropic log-concave measures, specifically $1/n$4 (Klartag 2023), and $1/n$5 bounds on the gradient of the logarithmic potential: $1/n$6 (Eldan–Klartag 2008). Both are essential to quantifying the sensitivity of measure under Minkowski perturbations.
3. Variational Approach: Using the second variation method (Kolesnikov–Milman), the exponent $1/n$7 for which power concavity holds is expressed in terms of Reilly-type integrals involving solutions to measure-weighted Poisson equations $1/n$8. The new energy estimate circumvents previous obstructions by globalizing the application of the Poincaré inequality, rather than restricting to convex sub-level sets of the potential gradient.
4. Regularization and Approximation: Convolution with Gaussian kernels is used to regularize log-concave measures, enabling passage of inequalities to general convex sets and arbitrary log-concave densities by weak convergence.

The key technical innovation lies in the new energy estimate (Lemma 3.1), which demonstrates that for every even isotropic log-concave probability measure $1/n$9, and the associated solution $n \geq 3$0,

$n \geq 3$1

uniformly. This is achieved by robustly controlling both the second-moment and the contribution from the gradient of the potential, exploiting symmetry and isotropic conditions.

Numerical and Theoretical Implications

The improvement to a $n \geq 3$2 exponent is sharp up to order of magnitude with respect to known obstacles arising from the first moment of $n \geq 3$3. The result closes the gap between the functional and geometric Brunn–Minkowski inequalities for large classes of measures and supports, and provides the strongest general upper bound toward the full $n \geq 3$4-concavity conjecture for even log-concave measures currently available.

Additionally, the methodology shows that further improvement, up to the conjectural $n \geq 3$5 bound, cannot follow by simply refining current techniques such as Poincaré inequalities or $n \geq 3$6 bounds on the potential gradient since these are known to be dimensionally optimal. A leap beyond $n \geq 3$7 would require fundamentally new analytic ideas or structural results about log-concave measures in high dimensions.

From a practical and theoretical viewpoint, this result has implications for:

• Geometric Analysis: Advances the understanding of isoperimetric-type and concavity inequalities for non-Euclidean measures, suggesting routes for tackling functional inequalities via analytic means.
• Information Theory and Probability: Provides new quantitative tools for studying concentration of measure and large deviations for convex sets under general high-dimensional distributions.
• Convex Geometry: Bridges the gap between combinatorial, geometric, and analytic approaches to extremal volume and surface area problems.

Potential Directions and Open Problems

While a significant advancement, several critical directions remain:

• Dimensional Improvement: Finding techniques to further improve the exponent toward $n \geq 3$8 concavity remains an outstanding challenge. Novel approaches beyond current analytic and variational methods are necessary.
• Extension to Non-Even Measures: The present result applies only to even log-concave measures. Establishing similar bounds without the symmetry assumption is a natural and challenging extension.
• Entropic and Functional Forms: The relation between concavity properties for measures, their marginals, and corresponding entropy/functional inequalities warrants deeper investigation, as highlighted in recent works cited by the paper.

Conclusion

This paper establishes a new uniform estimate for the power in the dimensional Brunn–Minkowski inequality for even log-concave measures, improving the exponent to $n \geq 3$9 and optimally refining previous bounds. The approach synthesizes analytic inequalities, measure regularization, and variational formulae, and sets a new benchmark for the concavity properties of high-dimensional log-concave measures. Achieving further dimensional improvement would require fundamentally new ideas, pointing to the depth and significance of the dimensional Brunn–Minkowski problem in modern geometric analysis.