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 \(B\subset \text {int} K\) such that for every support hyperplane \(\Pi \) of B, the section \(\Pi \cap K\) is a centrally symmetric set. In [4], it was conjectured that the Barker-Larman condition characterizes the ellipsoid. In this work, we prove a special case of such a 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 \(B\subset \operatorname {int}K\) be a ball 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 \(\operatorname {bd}B\) is not very close to \(\operatorname {bd}K\) , see the definition in the Introduction), then K is an ellipsoid.
