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.
A note on Bourgain’s slicing problem
3 Pith papers cite this work. Polarity classification is still indexing.
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Even log-concave measures satisfy μ(λK + (1-λ)L)^{c_n} ≥ λ μ(K)^{c_n} + (1-λ) μ(L)^{c_n} with c_n ≥ c/(n^3 ln n).
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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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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Functional perimeter and the dimensional Brunn-Minkowski inequality for log-concave measures
Even log-concave measures satisfy μ(λK + (1-λ)L)^{c_n} ≥ λ μ(K)^{c_n} + (1-λ) μ(L)^{c_n} with c_n ≥ c/(n^3 ln n).
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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.