An example related to the slicing inequality for general measures

For n∈N, let S_n 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 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/log⁡log⁡n, 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.