Distance inequalities of generalized k-intersection bodies

It was shown in [11] that for every symmetric star body K ⊆ Rⁿ of volume $1$, every even continuous probability density f on K and 1 ≤ k ≤ n-1, there exists a subspace F ⊆ Rⁿ of codimension k such that \[ ∫K ∩ F f ≥ c^k (dₒᵥᵣ(K, BP_k^n))⁻ᵏ \] where dₒᵥᵣ(K, BPₖⁿ) is the outer volume ratio distance from K to the class of generalized k-intersection bodies, and $c>0$ is a universal constant. The upper bound dₒᵥᵣ(K, BPₖⁿ) ≤ c' √n/k (log(enk))3/2 was established in [13] for every convex body K. In this note we show that there exist a symmetric convex body K of volume $1$ and an even continuous probability density f supported on K such that \[ ∫K ∩ F f ≤ ( c √{ n{k log(n)} } )⁻ᵏ. \] As a consequence we obtain a lower bound for dₒᵥᵣ(K, BPₖⁿ) with K a convex body, complementing the upper bound in [13]. This is \[c √n/k (log(n))-1/2 ≤ _K dₒᵥᵣ(K, BP_k^n) ≤ c' √n/k (log({en}k))3/2.\] The case $k=1$ was obtained previously in [5,6].