Concentration of mass and central limit properties of isotropic convex bodies Conference Paper uri icon

abstract

  • We discuss the following question: Do there exist an absolute constant c > 0 and a sequence φ(n) tending to infinity with n, such that for every isotropic convex body K in ℝn and every t ≥ 1 the inequality Prob ({x ∈ K: ∥x∥2 ≥ c√nLKt}) ≤ exp ( - φ(n)t) holds true? Under the additional assumption that K is 1-unconditional, Bobkov and Nazarov have proved that this is true with φ(n) ≃ √n. The question is related to the central limit properties of isotropic convex bodies. Consider the spherical average fK(t) = ∫Sn-1 |K ∩ (θ⊥ + tθ)|σ(dθ). We prove that for every γ ≥ 1 and every isotropic convex body K in ℝn, the statements (A) "for every t ≥ 1, Prob ({x ∈ K: ∥x∥2 ≥ γ√nLKt}) ≤ exp ( - φ(n)t)" and (B) "for every 0 < t ≤ c1(γ)√φ(n)LK, f K(t) ≤ c2/LK exp ( - t2/(c 3(γ)2LK2)), where c i(γ) ≃ γ" are equivalent.

author list (cited authors)

  • Paouris, G.

citation count

  • 7

publication date

  • September 2004