Suppose x and s lie in the interiors of a cone K and its dual K^* respectively. We seek dual ellipsoidal norms such that the product of the radii of the largest inscribed balls centered at x and s and incribed in K and K^* respectively is maximized. Here the balls are defined using the two dual norms. We provide a solution when the cones are symmetric, that is self-dual and homogeneous. This provides a geometric justification for the Nesterov-Todd primal-dual scaling in symmetric cone programming.
Technical Report No. 1426, School of Operations Research and Industrial Engineering, Cornell University