Todd's maximum-volume ellipsoid problem on symmetric cones
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Yongdo Lim
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.
© de Gruyter 2011
Articles in the same Issue
- On the ampleness of the normal bundle of line congruences
- Topological types of 3-dimensional small covers
- Null controllability with constraints on the state for nonlinear heat equations
- Exponential closing property and approximation of Lyapunov exponents of linear cocycles
- Reflection systems and partial root systems
- Todd's maximum-volume ellipsoid problem on symmetric cones
- Refinement of the spectral asymptotics of generalized Krein Feller operators
Articles in the same Issue
- On the ampleness of the normal bundle of line congruences
- Topological types of 3-dimensional small covers
- Null controllability with constraints on the state for nonlinear heat equations
- Exponential closing property and approximation of Lyapunov exponents of linear cocycles
- Reflection systems and partial root systems
- Todd's maximum-volume ellipsoid problem on symmetric cones
- Refinement of the spectral asymptotics of generalized Krein Feller operators
