Abstract
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.
Original language | English |
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Pages (from-to) | 2089-2112 |
Number of pages | 24 |
Journal | Journal of Functional Analysis |
Volume | 274 |
Issue number | 7 |
Early online date | 6 Sep 2017 |
DOIs | |
State | Published - 1 Apr 2018 |
All Science Journal Classification (ASJC) codes
- Analysis