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Math-2639 · mathematics · Geometry

Logarithmic Sobolev inequality from subgaussian marginals

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Does an absolute C exist such that each centered log-concave probability measure $\mu $ with a Lebesgue density on $R^{n}$ and each $a>0$ satisfying $sup_{|\theta |=1}\int \exp (\langle x,\theta \rangle ^{2}/a^{2})d\mu \leq 2$ obeys $Ent_\mu (f^{2})\leq Ca^{2}\int |\nabla f|^{2}d\mu $ for every compactly supported smooth f? $Ent_\mu (g)=\int g$ $logg-(\int g)\log (\int g)$. The same C must work for all $n,a,\mu $.

Short English statement

Does an absolute C exist such that each centered log-concave probability measure $\mu $ with a Lebesgue density on $R^{n}$ and each $a>0$ satisfying $sup_{|\theta |=1}\int \exp (\langle x,\theta \rangle ^{2}/a^{2})d\mu \leq 2$ obeys $Ent_\mu (f^{2})\leq Ca^{2}\int |\nabla f|^{2}d\mu $ for every compactly supported smooth f? $Ent_\mu (g)=\int g$ $logg-(\int g)\log (\int g)$. The same C must work for all $n,a,\mu $.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2639-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2639

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 093; 2026-09-23).

Logarithmic Sobolev inequality from subgaussian marginals

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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