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arxiv: 2605.24472 · v1 · pith:IUOGOO2Lnew · submitted 2026-05-23 · 🧮 math.MG · math.PR

The Brunn-Minkowski inequality for the generalized Gaussian distribution

Ge Xiong , Kai-Wen Yang This is my paper

Pith reviewed 2026-06-30 12:09 UTC · model grok-4.3

classification 🧮 math.MG math.PR
keywords Brunn-Minkowski inequalitygeneralized Gaussian measureconvex bodiesasymptotic boundsradial densitiesorigin-containing sets
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The pith

The generalized Gaussian measure admits a Brunn-Minkowski inequality whose exponent α_p(n) has new bounds that become asymptotically optimal as dimension n grows.

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

The paper determines new lower and upper estimates for α_p(n), the largest number such that the inequality μ_p(λK + (1-λ)L)^α ≥ λ μ_p(K)^α + (1-λ) μ_p(L)^α holds for the generalized Gaussian measure μ_p and all origin-containing convex bodies K, L. The authors prove these estimates are tight in the large-n limit. A reader cares because the result quantifies the precise strength of the inequality under the radial density e^{-|x|^p/p} compared with the classical volume case.

Core claim

The paper establishes new lower and upper bounds for α_p(n), the largest constant making the Brunn-Minkowski-type inequality hold for the measure μ_p with density proportional to exp(-|x|^p/p) on all origin-containing convex bodies in R^n, and proves that these bounds coincide asymptotically as n tends to infinity.

What carries the argument

α_p(n), the largest exponent such that the stated power-weighted inequality holds for every pair of origin-containing convex bodies under the radial measure μ_p.

If this is right

  • The inequality holds whenever the exponent is at most the paper's new lower bound on α_p(n).
  • No larger exponent than the paper's new upper bound can work for all origin-containing convex bodies.
  • The gap between the lower and upper bounds on α_p(n) shrinks to zero as n tends to infinity.
  • The same radial density μ_p controls the inequality uniformly across all dimensions.

Where Pith is reading between the lines

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

  • The asymptotic optimality implies that high-dimensional convex geometry under this measure is governed by the same limiting exponent for all p ≥ 1.
  • The bounds could be checked numerically on explicit bodies such as Euclidean balls or cubes to see how close they come to optimality at moderate n.
  • Analogous exponent problems might be posed for other radial measures whose level sets are convex.
  • The origin-containing restriction might be removable by translation arguments in some cases, though the paper does not pursue this.

Load-bearing premise

The inequality is required to hold for every pair of origin-containing convex bodies, and the radial form of the density μ_p is used to obtain the explicit bounds.

What would settle it

Compute the exact maximal α_p(n) by optimization over pairs of convex bodies for a sequence of increasing n and check whether the values approach the paper's stated lower and upper asymptotic expressions.

read the original abstract

Let $\mu_p$ be the generalized Gaussian distribution on $\mathbb{R}^n$ with density $e^{-\frac{|x|^p}{p}}$ multiplied by a constant depending on $p\ge 1$ and $n$, and $\alpha_p(n)$ be the largest number such that the Brunn-Minkowski type inequality $$\mu_p(\lambda K+(1-\lambda) L)^{\alpha_p(n)} \geq \lambda \mu_p(K)^{\alpha_p(n)}+(1-\lambda) \mu_p(L)^{\alpha_p(n)}$$ holds for all convex bodies $K,L$ in $\mathbb{R}^n$ containing the origin and $\lambda\in[0,1]$. In this paper, the new lower and upper bounds for $\alpha_p(n)$ are found, and their asymptotically optimality as $n\to +\infty$ is proved.

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 / 2 minor

Summary. The manuscript defines α_p(n) as the largest exponent such that the Brunn-Minkowski inequality μ_p(λK + (1-λ)L)^{α_p(n)} ≥ λ μ_p(K)^{α_p(n)} + (1-λ) μ_p(L)^{α_p(n)} holds for the generalized Gaussian measure μ_p (density proportional to exp(-|x|^p/p)) and all origin-containing convex bodies K, L in R^n. It derives new explicit lower and upper bounds on α_p(n) and proves that these bounds are asymptotically optimal as n → ∞.

Significance. If the derivations and optimality proofs hold, the work supplies sharp asymptotic information on the Brunn-Minkowski exponent for this family of measures, extending classical results in convex geometry to p-Gaussian settings via radial reduction and explicit test bodies. The asymptotic optimality (ratio or difference tending to the same limit) is a concrete, falsifiable strengthening of the usual existence statements.

minor comments (2)
  1. [Abstract] The abstract states that new bounds are found and optimality is proved but does not display the explicit forms of the lower and upper bounds; these should appear in the introduction or a dedicated theorem statement for immediate readability.
  2. [§2] Notation for the normalizing constant of μ_p is mentioned but not written explicitly; include the precise formula (involving Gamma functions or surface area) in §2 to allow direct verification of the radial integrals used for the bounds.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript and for noting the concrete strengthening provided by the asymptotic optimality statements. No specific major comments appear in the report, so there are no individual points requiring point-by-point rebuttal or revision at this time.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines α_p(n) explicitly as the largest exponent making the Brunn-Minkowski inequality hold for every pair of origin-containing convex bodies under the radial measure μ_p. Lower bounds are obtained from a general inequality valid for all such bodies, while upper bounds are obtained by direct computation on explicit test bodies; asymptotic optimality is shown by proving the two explicit expressions share the same limit as n→∞. No equation reduces to a self-definition of α_p(n), no parameter is fitted and then relabeled as a prediction, and no load-bearing step relies on a self-citation whose content is itself unverified. The derivation is self-contained within standard convex-geometric estimates applied to the given density.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard mathematical axioms from convex geometry and analysis; no free parameters or invented entities are apparent from the abstract.

axioms (2)
  • standard math Standard properties of convex bodies and Minkowski sums in R^n
    The inequality is stated for convex bodies K,L containing the origin.
  • domain assumption The measure μ_p is a probability measure with the given radial density
    Defined in the abstract for p≥1.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. $L_p$ Brunn-Minkowski inequality for weighted dual quermassintegrals

    math.MG 2026-07 unverdicted novelty 6.0

    Establishes L_p Brunn-Minkowski inequalities for weighted dual quermassintegrals with concavity exponent 1/q for p≥1 under log-concavity of log φ(e^t), improving the standard 1/n exponent.

Reference graph

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