Bounding the Norm of a Log-Concave Vector Via Thin-Shell Estimates

Ronen Eldan (1) , Joseph Lehec (2)

1 University of Washington [Seattle]
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

Chaining techniques show that if X is an isotropic log-concave random vector in R n and Γ is a standard Gaussian vector then EX ≤ Cn 1/4 EΓ for any norm · , where C is a universal constant. Using a completely different argument we establish a similar inequality relying on the thin-shell constant σn = sup Var(|X|); X isotropic and log-concave on R n . In particular, we show that if the thin-shell conjecture σn = O(1) holds, then n 1/4 can be replaced by log(n) in the inequality. As a consequence, we obtain certain bounds for the mean-width, the dual mean-width and the isotropic constant of an isotropic convex body. In particular, we give an alternative proof of the fact that a positive answer to the thin-shell conjecture implies a positive answer to the slicing problem, up to a loga-rithmic factor.

Domaines: Mathématiques [math]

Ronen Eldan, Joseph Lehec. Bounding the Norm of a Log-Concave Vector Via Thin-Shell Estimates. Bo'az Klartag; Emanuel Milman. Geometric Aspect of Functional Analysis, 2116, Springer, pp.107 - 122, 2014, Lecture Notes in Mathematics, ⟨10.1007/978-3-319-09477-9_9⟩. ⟨hal-01100946⟩