I wanted to ask if the following holds for every convex centered body $K$ (in $\mathbb{R}^{n}$):
$\int_{S^{n-1}}|K\cap{\theta^{\perp}|}d\theta\geq|K|$
where $|K|$ is the $n$ volume of K and $|K\cap{\theta^{\perp}}|$ is the $n-1$ volume of $K\cap{\theta^{\perp}}$.
Thanks.
The claim is wrong. One can convince himself that the euclidian ball is a counter example when $n$ is large.