  It was shown in [11] that for every origin-symmetric star body $K \subseteq
\mathbb R^n$ of volume $1$, every even continuous probability density $f$ on
$K$ and $1 \leq k \leq n-1$, there exists a subspace $F \subseteq \mathbb R^n$
of codimension $k$ such that
  \[
  \int_{K \cap F} f \geq c^k (d_{\rm ovr}(K, \mathcal{BP}_k^n))^{-k}
  \]
  where $d_{\rm ovr}(K, \mathcal{BP}_k^n)$ 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 $d_{\rm ovr}(K, \mathcal{BP}_k^n) \leq c' \sqrt{n/k}
\left(\log\left(\frac{en}k\right)\right)^{3/2}$ was established in [13] for
every 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 $k$,
  \[
  \int_{K \cap F} f \leq \left( c \sqrt{\frac n{k \log(n)} } \right)^{-k}.
  \]
  As a consequence we obtain a lower bound for $d_{\rm ovr}(K,
\mathcal{BP}_k^n)$ with $K$ a convex body, complementing the upper bound in
[13]. This is \[c \sqrt{n/k} (\log(n))^{-1/2} \leq \sup_K d_{\rm ovr}(K,
\mathcal{BP}_k^n) \leq c' \sqrt{n/k}
\left(\log\left(\frac{en}k\right)\right)^{3/2}.\]
  The case $k=1$ was obtained previously in [5,6].
