Non-central sections of convex bodies

We study the following open problem, suggested by Barker and Larman. Let $K$ and $L$ be convex bodies in $\mathbb R^n$ ($n\ge 2$) that contain a Euclidean ball $B$ in their interiors. If $\mathrm{vol}_{n-1}(K\cap H) = \mathrm{vol}_{n-1}(L\cap H)$ for every hyperplane $H$ that supports $B$, does it follow that $K=L$? We discuss various modifications of this problem. In particular, we show that in $\mathbb R^2$ the answer is positive if the above condition is true for two disks, none of which is contained in the other. We also study some higher dimensional analogues.

No reviews yet.

No comments yet.