Approximation of convex bodies by centrally symmetric bodies

We present an analog of the well-known theorem of F. John about the ellipsoid of maximal volume contained in a convex body. Let C be a convex body and let D be a centrally symmetric convex body in the Euclidean d-space. We prove that if D-l is an affine image of D of maximal possible volume contained in C, then C a subset of the homothetic copy of D-l with the ratio 2d - 1 and the homothety center in the center of D-l. The ratio 2d - 1 cannot be lessened as a simple example shows.