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Math-2386 · mathematics · Probability

Variance Conjecture for Log-Concave Random Vectors

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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.
Does there exist an absolute constant $C$ such that every centered log-concave random vector $X\in\mathbb R^n$ satisfies $\operatorname{Var}(|X|^2)\leq C\lambda_X^2\mathbb E|X|^2$, where $\lambda_X^2$ is the largest eigenvalue of its covariance matrix?

Formulation references: [S1], [S2].

Short English statement

Does there exist an absolute constant $C$ such that every centered log-concave random vector $X\in\mathbb R^n$ satisfies $\operatorname{Var}(|X|^2)\leq C\lambda_X^2\mathbb E|X|^2$, where $\lambda_X^2$ is the largest eigenvalue of its covariance matrix?

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2386-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2386

The logarithmic Sobolev conclusion assumes a uniform subgaussian bound on every linear marginal. General log-concave laws need not have that bound at a covariance-comparable scale, so the unrestricted variance or KLS assertion here does not follow.

Variance Conjecture for Log-Concave Random Vectors

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 093; 2026-09-23), the asserted result is that a centered log-concave density with linear subgaussian parameter $a$ satisfies a logarithmic Sobolev inequality with constant bounded by an absolute multiple of $a^2$.

Mathematical statement of the claimed result

Manuscript 1.
Take any $n\ge1$, a centered log-concave probability measure $\mu$ on $\mathbb R^n$ with Lebesgue density, and $a>0$ such that $X\sim\mu$ obeys \[ \sup_{|\theta|=1}\mathbb E\exp(\langle X,\theta\rangle^2/a^2)\le2. \] The claimed absolute, dimension-independent constant $C$ gives, for all $f\in C_c^{\infty}(\mathbb R^n)$, \[ \operatorname{Ent}_{\mu}(f^2)\le Ca^2\int|\nabla f|^2\,d\mu, \qquad \operatorname{Ent}_{\mu}(g)=\int g\log g\,d\mu- (\int g\,d\mu)\log(\int g\,d\mu). \] Only linear-function subgaussianity is assumed, but a logarithmic Sobolev conclusion is claimed.

Outline of the argument

For manuscript 1 (family 093; 2026-09-23), gaussian regularization and entropy dissipation reduce the question to critical posterior laws. Efficient observation probes and nearly-first-eigenfunction hierarchies control the prediction curve, while curvature and operator-tail bounds rule out a dimension-dependent obstruction.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 093; 2026-09-23).

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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