Maximum-volume symmetric gauge ball problem on the convex cone of positive definite matrices and convexity of optimal sets

In this paper we show that the radius of the largest symmetric gauge ball inscribed in the convex cone PD(n) of n×n positive definite matrices equipped with the Finsler metric inherited from a unitarily invariant norm is precisely the minimal eigenvalue of its center. We then solve Todd's largest dual ellipsoids problem [Math. Program. Ser. B, 117 (2009), pp. 425-434] for unitarily invariant norms, i.e., the problem of maximizing the product of the unitarily invariant norm distances to boundaries of the cone and its dual cone. We further show that the optimal set of maximizers forms a convex cone and is a closed geodesically convex subset of the Riemannian manifold PD(n). This in particular provides a one-parameter family of strictly increasing solid convex (in both a Euclidean and a Riemannian sense) cones of positive definite matrices which starts from the optimal set and covers PD(n).