On asphericity of convex body

The paper deals with a finite-dimensional problem of minimizing for a given convex body the ratio of the circumscribed ball radius to the inscribed ball radius (in an arbitrary norm) by choosing the common center of these balls. We establish quasiconvexity and subdifferentiability of the objective function of this problem. We find a criterion of a solution and conditions of its uniqueness. The main problem is compared with problems which are close to it in geometric sense.