It was shown in [11] that for every origin-symmetric star body K ⊆ ℝn of volume 1, every even continuous probability density f on K and \(1\le k\le n-1,\) there exists a subspace F ⊆ ℝn of codimension k such that \(\int_{K\cap F}f\ge c^{k}(d_{\text{ovr}}(K,{\cal{B}}{\cal{P}}_{k}^{n}))^{-k}\) where \({d_{{\text{ovr}}}}({K,{\cal{B}}{\cal{P}}_k^n})\) 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 \({d_{{\text{ovr}}}}({K,{\cal{B}}{\cal{P}}_k^n}) \le c^{\prime}\sqrt {n/k}{\left({\log \left({{{en} \over k}} \right)} \right)^{3/2}}\) was established in [13] for every origin-symmetric convex body K. In this note we show that there exist an origin-symmetric convex body K of volume 1 and an even continuous probability density f supported on K such that for every subspace F of codimension k \(\int_{K\cap F}f\le \left(c\sqrt{{n \over {k\log n}}}\right)^{-k}.\)
As a consequence we obtain a lower bound for \({d_{{\text{ovr}}}}({K,{\cal{B}}{\cal{P}}_k^n})\) with K a convex body, complementing the upper bound in [13]. This is
\(c\sqrt {n/k} {({\log (n)})^{- 1/2}} \le \mathop {\sup}\limits_K {d_{{\text{ovr}}}}({K,{\cal{B}}{\cal{P}}_k^n}) \le c^{\prime}\sqrt {n/k}{\left({\log \left({{{en} \over k}} \right)} \right)^{3/2}}\)
The case k = 1 was obtained previously in [5, 6].
