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 \ ₊ ₅ f cᵏ (d ₎ₕₑ (K, BPₖⁿ) ) ^-k \ where d ₎ₕₑ (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 d ₎ₕₑ (K, BPₖⁿ) c' n/k ( (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 \ ₊ ₅ f (c nk (n) ) ^-k. \ As a consequence we obtain a lower bound for d ₎ₕₑ (K, BPₖⁿ) with K a convex body, complementing the upper bound in 13. This is n/k ( (n) ) ^-1/2 K d ₎ₕₑ (K, BPₖⁿ) c' n/k ( (enk) ) ^3/2. \ The case k=1 was obtained previously in 5, 6.