Convex bodies with centrally symmetric sections

Let $K\subset {\mathbb{R}}^n$ be a convex body, $n\ge 3$. We say that $K$ satisfies the Barker-Larman condition if there exists a ball $B$ in the interior of $K$ such that for every suppor hyperplane $\mathrm{\Pi }$ of $B$, the section $\mathrm{\Pi }\cap K$ is a centrally symmetric set. Barker and Larman conjectured that the Barker-Larman condition characterizes the ellipsoid. In this work we prove an special case of such conjecture, in particular, we assume that the convex body $K$ is centrally symmetric. Our main result is the following: Let $K$ be a centrally symmetric and strictly convex body, with center at $O$, and let $B$ be a ball in the interior of $K$ and not containing $O$: If $K$ satisfies the Barker-Larman condition with respect to $B$ and $B$ is suitable for $K$ (intuitively, $B$ is suitable for $K$ if the boundary of $B$ is not very close to the boundary of $K$), then $K$ is an ellipsoid.