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The slicing conjecture via small ball estimates

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arxiv 2501.06854 v1 pith:JWJ6XVDT submitted 2025-01-12 math.FA math.PR

The slicing conjecture via small ball estimates

Pierre Bizeul This is my paper
classification math.FA math.PR
keywords ballconjectureslicingsmallestimatesklartaglehecalternative
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved
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Bourgain's slicing conjecture was recently resolved by Joseph Lehec and Bo'az Klartag. We present an alternative proof by establishing small ball probability estimates for isotropic log-concave measures. Our approach relies on the stochastic localization process and Guan's bound, techniques also used by Klartag and Lehec. The link between small ball probabilities and the slicing conjecture was first observed by Dafnis and Paouris and is established through Milman's theory of M-ellipsoids.

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Cited by 5 Pith papers

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  1. The KLS constant is $O(\log^{1/4} n)$

    math.PR 2026-07 conditional novelty 7.0

    Every isotropic log-concave measure satisfies a quadratic-form Poincaré inequality with constant 2, which implies the KLS constant is at most C log^{1/4} n.

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  4. Minimum Norm Interpolation via The Local Theory of Banach Spaces: The Role of Gaussianity

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    The sharp MSE bound for the ℓ1-minimum-norm interpolator under isotropic Gaussian covariates is recovered via the geometry of symmetric Gaussian polytopes, without the convex Gaussian min-max theorem.

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    math.MG 2026-05 unverdicted novelty 6.0

    Proves dimensional Brunn-Minkowski inequality for even log-concave measures with c_n ≥ c/(n^3 ln n) and shows Γ_n ≈ n for maximal functional perimeter of isotropic log-concave measures.