On parallelepipeds of minimal volume containing a convex symmetric body in ℝn

Abstract Given a convex symmetric body C ⊂ ℝ n we put a (C) = | C | sup | P | −1 where the supremum extends over all parallelepipeds containing C and | A | denotes the volume of a set A ⊂ ℝ n. Let a n = inf a (C): C ⊂ ℝ n. We show that which slightly improves the estimate due to Dvoretzky and Rogers 6. In every dimension n we construct a convex symmetric polytope W n such that the unit Euclidean ball is the ellipsoid of maximal volume inscribed into W n and the volume of every parallelepiped containing W n is greater than for large n which shows ‘the limit’ to the Dvoretzky Rogers method for bounding a n below. We present an alternative proof of the result of I. K. Babenko l that. We show that, and that a local minimum of the function C → a (C) for C ⊂ ℝ n is attained only at an equiframed convex body (that is, a body such that every point of its boundary belongs to a parallelepiped of minimal volume containing the body).