Application of an idea of Voronoĭ to John type problems

Using an idea of Voronoĭ, many John type and minimum position problems in dimension d can be transformed into more accessible geometric problems on convex subsets of the -dimensional cone of positive definite quadratic forms. In this way, we prove several new John type and minimum position results and give alternative versions and extensions of known results. In particular, we characterize minimum ellipsoidal shells of convex bodies and, in the typical case, show their uniqueness and determine the contact number. These results are formulated also in terms of the circumradius of convex bodies. Next, circumscribed ellipsoids of minimum surface area of a convex body and the corresponding minimum position problem are studied. Then we investigate John type characterizations of minimum positions of a convex body with respect to moments and the product of a moment and the moment of the polar body. The technique used in this context, finally, is applied to obtain corresponding results for the mean width and the surface area.