Symmetric Cones

Abstract

Joe D’Arti was an ever great source for open problems. One of the questions he enjoyed was the following: Let K denote an open convex cone in ℝⁿ, with vertex at the origin, that does not contain any straight line. Then K✶ = {y ∈ ℝn: 〈y,x〉 > 0 for all x ∈ K \ {0}} denotes the dual cone for K, where 〈·,·〉 is the standard inner product on ℝn and K denotes the closure in ℝn of K. Consider the function

$$\eta (x) = \int\limits_K {{e^{ - \left\langle {y,x} \right\rangle }}dy.}$$

. This function is defined for all x ∈ K and real analytic [2, §3]. Moreover η is “logarithmica;;y convex”, i.e., g _x (u, v) = (d ² _x ln η)(u, v) is, for every x ∈ K, a positive definite symmetric bilinear form on ℝⁿ [2, §3.5]. In other words, (g _x ) _x∈K defines a Riemannian metric on K.

Supported by NSF Grant DMS-9205293 and the Deutsche Forschungsgemeinschaft.