Dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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Dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures

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Dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures. Proves a dimension-free logarithmic Sobolev inequality for centered log-concave densities with uniformly subgaussian linear marginals, with constant bounded by a universal multiple of the squared linear subgaussian parameter.

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LEVEL 1 A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures

released 2026-09-23 | 1 theorem · 35 lemmas · 44 proofs · 26,206 words | PLAY LEVEL 1 » (pdf)

We prove that every centered log-concave probability measure with a Lebesgue density on ℝⁿ and linear subgaussian parameter a satisfies $\mathop{\mathrm{Ent}}\nolimits _\mu(f^2)\le Ca^2\int|Df|^2\,d\mu$ for compactly supported smooth f, with one universal constant C. This resolves positively the dimension-free logarithmic Sobolev conjecture for subgaussian log-concave measures.

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