Osaka Journal of Mathematics

Metric convexity of symmetric cones

Jimmie Lawson and Yongdo Lim

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In this paper we introduce a general notion of a symmetric cone, valid for the finite and infinite dimensional case, and prove that one can deduce the seminegative curvature of the Thompson part metric in this general setting, along with standard inequalities familiar from operator theory. As a special case, we prove that every symmetric cone from a JB-algebra satisfies a certain convexity property for the Thompson part metric: the distance function between points evolving in time on two geodesics is a convex function. This provides an affirmative answer to a question of Neeb [22].

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Osaka J. Math., Volume 44, Number 4 (2007), 795-816.

First available in Project Euclid: 7 January 2008

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Primary: 53C35: Symmetric spaces [See also 32M15, 57T15] 47H07: Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces 46B20: Geometry and structure of normed linear spaces 53B40: Finsler spaces and generalizations (areal metrics)


Lawson, Jimmie; Lim, Yongdo. Metric convexity of symmetric cones. Osaka J. Math. 44 (2007), no. 4, 795--816.

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