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Math-2640 · mathematics · Geometry

Sharp entropy lower bound for log-concave densities

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every log-concave probability density f on $R^{m}$, must its differential entropy $h(f)=-\int f$ logf satisfy $h(f)\geq m+\tfrac12 \log $ det Cov(f)? Is equality equivalent, up to almost-everywhere change, to f being the density of $M(E_{1},\dots ,E_m)+b$ with M invertible and independent standard exponential $E_i$ supported on $[0,\infty )$? No symmetry or smoothness is assumed.

Short English statement

For every log-concave probability density f on $R^{m}$, must its differential entropy $h(f)=-\int f$ logf satisfy $h(f)\geq m+\tfrac12 \log $ det Cov(f)? Is equality equivalent, up to almost-everywhere change, to f being the density of $M(E_{1},\dots ,E_m)+b$ with M invertible and independent standard exponential $E_i$ supported on $[0,\infty )$? No symmetry or smoothness is assumed.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2640-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2640

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 101; 2026-10-05).

Sharp entropy lower bound for log-concave densities

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 101; 2026-10-05), the asserted result is that simplices uniquely maximize the isotropic constant, and log-concave densities satisfy the stated sharp entropy--covariance bound with affine exponential-product equality.

Mathematical statement of the claimed result

Manuscript 1.
For every $n\ge1$ and every full-dimensional convex body $K\subset\mathbb R^n$, use $V=|K|$, its centroid $b$, covariance $\Sigma_K=V^{-1}\int_K(x-b)(x-b)^{\mathsf T}\,dx$, and $L_K=(\det\Sigma_K/V^2)^{1/(2n)}$. The claimed sharp bound, with equality exactly for simplices, is \[ L_K\le\frac{(n!)^{1/n}}{(n+1)^{(n+1)/(2n)}\sqrt{n+2}}. \] The associated entropy assertion concerns every log-concave probability density $f$ on $\mathbb R^m$, $m\ge1$: \[ h(f)=-\int f\log f\ge m+\tfrac12\log\det\operatorname{Cov}(f). \] Equality is claimed precisely for affine images $M(E_1,\ldots,E_m)^{\mathsf T}+b$, with $M$ invertible and independent standard exponential variables $E_i$; densities are identified almost everywhere.

Outline of the argument

For manuscript 1 (family 101; 2026-10-05), gaussian decomposition expands the entropy deficit into weighted chaos terms and commutator squares. Certified scalar inequalities control the expansion, and vanishing slack forces an exponential-product law; applying the bound to uniform bodies yields simplex rigidity.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 101; 2026-10-05).

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

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