## Metric convexity of symmetric cones.(English)Zbl 1135.53014

Using an algebraic construction of symmetric sets, the authors 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. A special case is the symmetric cone arising as the set of invertible squares of a Jordan-Banach algebra (JB-algebra). The authors 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 K.-H. Neeb [Geom. Dedicata 95, 115–156 (2002; Zbl 1027.58003)].

### MSC:

 53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics) 53C35 Differential geometry of symmetric spaces 46H70 Nonassociative topological algebras 46B20 Geometry and structure of normed linear spaces

Zbl 1027.58003
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