Let $K$ be a convex centered body in $\mathbb{R}^{n}$ and suppose that for every $\theta\in{S^{n-1}}$ we have $|K\cap{\theta^{\perp}}|=C$. Does this imply that $K$ is the euclidian ball?
One can consider a refinement of the first version, assuming moreover that all such sections are identical. Is it now the euclidian ball?
BTW, centered means that $\int_{K}xdx=0$
Thanks.