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Licensed Unlicensed Requires Authentication Published by De Gruyter April 2, 2011

Todd's maximum-volume ellipsoid problem on symmetric cones

Yongdo Lim
From the journal

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.

Received: 2009-05-06
Revised: 2009-06-10
Published Online: 2011-04-02
Published in Print: 2011-March

© de Gruyter 2011