An example related to the slicing inequality for general measures

For n∈N, let Sn be the smallest number S>0 satisfying the inequality∫Kf≤S⋅|K|1n⋅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/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 ψα-condition.