Todd's maximum-volume ellipsoid problem on symmetric cones

Abstract

Let V be a Euclidean Jordan algebra and let Ω be the associated symmetric cone, a self-dual homogeneous open convex cone, which is a symmetric space of noncompact type under G(Ω) (the linear automorphism group)-invariant Riemannian metric. We show that the radius of the largest ball centered at a ∈ Ω inscribed in Ω coincides with its minimum eigenvalue and then provide a proof of the problem of finding a point x ∈ Ω to maximize the product of the radii of the largest balls centered at a, b ∈ Ω and inscribed in Ω of the tangent space Tx (Ω) and its dual space Tx–1 (Ω), respectively. We obtain an explicit formula for the maxima; it is precisely the minimal eigenvalue of P (a1/2)b where P denotes the quadratic representation of V. This provides an affirmative answer to a question of Todd on the maxima.