About: Busemann–Petty problem

In the mathematical field of convex geometry, the Busemann–Petty problem, introduced by Herbert Busemann and , asks whether it is true that a symmetric convex body with larger central hyperplane sections has larger volume. More precisely, if K, T are symmetric convex bodies in Rn such that for every hyperplane A passing through the origin, is it true that Voln K ≤ Voln T? Busemann and Petty showed that the answer is positive if K is a ball. In general, the answer is positive in dimensions at most 4, and negative in dimensions at least 5.