TY - JOUR

T1 - An example related to the slicing inequality for general measures

AU - Klartag, Bo'az

AU - Koldobsky, Alexander

N1 - Publisher Copyright:
© 2017 Elsevier Inc.

PY - 2018/4/1

Y1 - 2018/4/1

N2 - For n∈N, let Sn be the smallest number S>0 satisfying the inequality ∫Kf≤S⋅|K|[Formula presented]⋅maxξ∈Sn−1∫K∩ξ⊥f for all centrally-symmetric convex bodies K in Rn and all even, continuous probability densities f on K. Here |K| is the volume of K. It was proved in [16] that Sn≤2n, and in analogy with Bourgain's slicing problem, it was asked whether Sn is bounded from above by a universal constant. In this note we construct an example showing that Sn≥cn/loglogn, where c>0 is an absolute constant. Additionally, for any 0<α<2 we describe a related example that satisfies the so-called ψα-condition.

AB - For n∈N, let Sn be the smallest number S>0 satisfying the inequality ∫Kf≤S⋅|K|[Formula presented]⋅maxξ∈Sn−1∫K∩ξ⊥f for all centrally-symmetric convex bodies K in Rn and all even, continuous probability densities f on K. Here |K| is the volume of K. It was proved in [16] that Sn≤2n, and in analogy with Bourgain's slicing problem, it was asked whether Sn is bounded from above by a universal constant. In this note we construct an example showing that Sn≥cn/loglogn, where c>0 is an absolute constant. Additionally, for any 0<α<2 we describe a related example that satisfies the so-called ψα-condition.

UR - http://www.scopus.com/inward/record.url?scp=85028889689&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2017.08.025

DO - 10.1016/j.jfa.2017.08.025

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AN - SCOPUS:85028889689

SN - 0022-1236

VL - 274

SP - 2089

EP - 2112

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 7

ER -