Midpoints for Thompson's metric on symmetric cones

TL;DR

This paper characterizes the structure of midpoint sets for Thompson's metric on symmetric cones, providing explicit formulas for their affine span dimensions in simple Euclidean Jordan algebras, advancing geometric understanding in this area.

Contribution

It offers a new characterization of the affine span of midpoint sets for Thompson's metric on symmetric cones, linking it to Peirce decomposition and deriving explicit dimension formulas.

Findings

Explicit formula for the dimension of the affine span of midpoints.

Characterization of midpoint sets via Peirce decomposition.

Application to simple Euclidean Jordan algebras.

Abstract

We characterise the affine span of the midpoints sets for Thompson's metric on symmetric cones in terms of a translation of the zero-component of the Peirce decomposition of an idempotent. As a consequence we derive an explicit formula for the dimension of the affine span of the midpoints sets in case the associated Euclidean Jordan algebra is simple.