All radial log-concave maximal function L^p norms converge to a universal limit λ(p) as dimension→∞, with the Euclidean ball asymptotically extremal.
A note on Bourgain’s slicing problem
6 Pith papers cite this work. Polarity classification is still indexing.
6
Pith papers citing it
abstract
This note is to study Bourgain's slicing problem following the routes investigated in the last decade. We show that the slicing constant $L_n$ is bounded by $C\log(\log n) $, $n\geq 3$, for some universal constant $C$.
years
2026 6representative citing papers
Nearly-tight O(sqrt(d)) approximation for zonotope containment in the oracle model, with a proof of Talagrand's conjecture for constant Delta-modular zonotopes and a tight Theta(d/log d) bound for general convex bodies.
Random normed spaces from isotropic log-concave measures satisfy d_BM >= cn / ln(1+m/n) with high probability, sharp in both parameters and recovering the order-n extremal when m is linear in n.
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.
Establishes dimension- and step-optimal Wasserstein bounds for DDPMs under Lipschitz score conditions and broad variance schedules via Föllmer process analysis, recovering prior results and extending to log-concave targets.
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.
citing papers explorer
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High-dimensional limits and extremizers for maximal functions associated with log-concave densities
All radial log-concave maximal function L^p norms converge to a universal limit λ(p) as dimension→∞, with the Euclidean ball asymptotically extremal.
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Nearly-Tight Bounds for Zonotope Containment and Beyond
Nearly-tight O(sqrt(d)) approximation for zonotope containment in the oracle model, with a proof of Talagrand's conjecture for constant Delta-modular zonotopes and a tight Theta(d/log d) bound for general convex bodies.
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Banach-Mazur distances and basis constants of isotropic log-concave random spaces
Random normed spaces from isotropic log-concave measures satisfy d_BM >= cn / ln(1+m/n) with high probability, sharp in both parameters and recovering the order-n extremal when m is linear in n.
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Minimum Norm Interpolation via The Local Theory of Banach Spaces: The Role of Gaussianity
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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Wasserstein bounds for denoising diffusion probabilistic models via the F\"ollmer process
Establishes dimension- and step-optimal Wasserstein bounds for DDPMs under Lipschitz score conditions and broad variance schedules via Föllmer process analysis, recovering prior results and extending to log-concave targets.
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Functional perimeter and the dimensional Brunn-Minkowski inequality for log-concave measures
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.