#### Title

Metric convexity of symmetric cones

#### Document Type

Article

#### Publication Date

12-1-2007

#### Abstract

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 geodesies is a convex function. This provides an affirmative answer to a question of Neeb [22].

#### Publication Source (Journal or Book title)

Osaka Journal of Mathematics

#### First Page

795

#### Last Page

816

#### Recommended Citation

Lawson, J., & Lim, Y.
(2007). Metric convexity of symmetric cones.* Osaka Journal of Mathematics**, 44* (4), 795-816.
Retrieved from https://repository.lsu.edu/mathematics_pubs/607