  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.
