TY - JOUR

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

AU - Klartag, Bo'az

AU - Koldobsky, Alexander

N1 - Communicated by E. Milman. We would like to thank Sergey Bobkov for encouraging us to discuss and think on this problem. We thank the anonymous referee for a thoughtful report. A major part of the work was done when both authors were visiting the Banff International Research Station during May 21–26, 2017. We would like to thank BIRS for hospitality. The first-named author was supported in part by a European Research Council (ERC) grant 305926. The second-named author was supported in part by the US National Science Foundation grant DMS-1700036.

PY - 2018/4/1

Y1 - 2018/4/1

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

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

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

U2 - https://doi.org/10.1016/j.jfa.2017.08.025

DO - https://doi.org/10.1016/j.jfa.2017.08.025

M3 - مقالة

SN - 0022-1236

VL - 274

SP - 2089

EP - 2112

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 7

ER -