we-solve (beta)

Math-2284 · mathematics · Analysis

Log-Minkowski inequality for symmetric convex bodies

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Log-Minkowski inequality

Show that if $K$ and $L$ origin-symmetric convex bodies with nonempty interior in ${\mathbb{R}}^n$, then $$ \int\limits_{S^{n-1}} \log{\frac{h_K(u)}{h_L(u)}}\,d\bar{V}_L(u) \geq \frac1n \log{\frac{|K|}{|L|}}, $$ where $\bar{V}_L$ is the cone-volume probability measure of $L$.

Formulation references: [S1], [S2].

Short English statement

Log-Minkowski inequality

Show that if $K$ and $L$ origin-symmetric convex bodies with nonempty interior in ${\mathbb{R}}^n$, then $$ \int\limits_{S^{n-1}} \log{\frac{h_K(u)}{h_L(u)}}\,d\bar{V}_L(u) \geq \frac1n \log{\frac{|K|}{|L|}}, $$ where $\bar{V}_L$ is the cone-volume probability measure of $L$.

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2284-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2284

The question is the cone-volume integral inequality for origin-symmetric convex bodies. The source claims the logarithmic Brunn--Minkowski volume inequality for the same bodies in every dimension. Applying the cone-volume first-variation formula at $L$ to the logarithmic Wulff path yields precisely the displayed integral bound. The source also identifies the two formulations as equivalent. No tensorization or local-product condition occurs in this canonical question, so the claimed result covers it fully.

Log-Minkowski inequality for symmetric convex bodies

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 091; 2026-09-23), the asserted result is that symmetric convex bodies satisfy logarithmic Brunn--Minkowski in all dimensions, with the stated $L_p$, even-measure, and dilation consequences.

Mathematical statement of the claimed result

Manuscript 1.
For a full-dimensional origin-symmetric convex body $K\subset\mathbb R^n$, write $h_K(u)=\sup_{x\in K}x\cdot u$, and for a positive function $g$ on the unit sphere define $W[g]=\{x:x\cdot u\le g(u)\text{ for every unit }u\}$. For every integer $n\ge1$, every such pair $K,L$, and every $\lambda\in[0,1]$, the claim is \[ |W[h_K^{1-\lambda}h_L^{\lambda}]|\ge |K|^{1-\lambda}|L|^{\lambda}. \] Its stated consequence, for every $0<p<1$, is \[ \left|W[((1-\lambda)h_K^p+\lambda h_L^p)^{1/p}]\right|^{p/n} \ge(1-\lambda)|K|^{p/n}+\lambda|L|^{p/n}. \] Volumes are Lebesgue volumes, and central symmetry is an assumption in both assertions.
Consequence for the recorded question.
Assuming the claimed logarithmic volume inequality, let $K,L$ be origin-symmetric convex bodies and let $W_t$ have Wulff constraints $h_L(u)^{1-t}h_K(u)^t$. Cone-volume first variation gives \[ \left.\frac{d}{dt}\right|_{t=0+}\log |W_t| =n\int_{S^{n-1}}\log\!\left(\frac{h_K}{h_L}\right)\,d\bar V_L. \] The asserted lower volume bound then implies \[ \int_{S^{n-1}}\log\!\left(\frac{h_K}{h_L}\right)\,d\bar V_L \ge\frac1n\log\!\left(\frac{|K|}{|L|}\right). \] This is a consequence of the selected claim, using the standard first-variation identity, rather than an additional theorem attributed to another paper.

Outline of the argument

For manuscript 1 (family 091; 2026-09-23), symmetric polytopes are approximated by smooth even-power weights. Logarithmic concavity of their integrals follows from a variance estimate in moment coordinates, and limits first remove the power smoothing and then the Gaussian regularization.

Formal verification

For manuscript 1 (family 091; 2026-09-23), OpenAI lists Lean statements for this assertion: symmetric convex bodies satisfy logarithmic Brunn--Minkowski in all dimensions, with the stated $L_p$, even-measure, and dilation consequences.

This coverage does not establish a formal proof of the complete problem as stated here.

The linked family Comparator checks are LogBrunnMinkowski.lean.

Their named declarations include OAI.LogBrunnMinkowski.main.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

Go to contribution →