Midpoints for Thompson's metric on symmetric cones

Gauge-reversing maps on cones, and Hilbert and Thompson isometries

We show that a cone admits a gauge-reversing map if and only if it is a symmetric cone. We use this to prove that every isometry of a Hilbert geometry is a collineation unless the Hilbert geometry is the projective space of a non-Lorentzian symmetric cone, in which case the collineation group is of index two in the isometry group. We also determine the isometry group of the Thompson geometry on a cone.

Symmetric cones, the Hilbert and Thompson metrics

Symmetric cones can be endowed with at least two interesting non Riemannian metrics: the Hilbert and the Thompson metrics. It is trivial that the linear maps preserving the cone are isometries for those two metrics. Oddly enough those are not the only isometries in general. We give here a full description of the isometry groups for both the Hilbert and the Thompson metrics using essentially the theory of euclidean Jordan algebras. Those results were already proved for the symmetric cone of complexe positive hermitian matrices by L. Moln{á}r. In this paper however we do not make any assumption …

JB-代数锥上的Hilbert和Thompson等距映射

Hilbert and Thompson isometries on cones in JB-algebras

本文刻画了JBW-代数和JB-代数锥内部的Hilbert与Thompson度量等距映射，推广了已有在对称锥及$C^*$-代数上的相关工作，并发展了新的几何与Jordan代数技巧。

Unique geodesics for Thompson's metric

In this paper a geometric characterization of the unique geodesics in Thompson's metric spaces is presented. This characterization is used to prove a variety of other geometric results. Firstly, it will be shown that there exists a unique Thompson's metric geodesic connecting $x$ and $y$ in the cone of positive self-adjoint elements in a unital $C^*$-algebra if, and only if, the spectrum of $x^{-1/2}yx^{-1/2}$ is contained in $\{1/β,β\}$ for some $β\geq 1$. A similar result will be established for symmetric cones. Secondly, it will be shown that if $C^\circ$ is the interior of a finite-dimensi…

A Metric Inequality for the Thompson and Hilbert Geometries

There are two natural metrics defined on an arbitrary convex cone: Thompson's part metric and Hilbert's projective metric. For both, we establish an inequality giving information about how far the metric is from being non-positively curved.

锥上保序流的Thompson度量收缩率——应用于广义Riccati方程

The contraction rate in Thompson metric of order-preserving flows on a cone - application to generalized Riccati equations

本文给出了闭凸尖锥内部保序流在Thompson部分度量下的Lipschitz常数公式，并应用于随机线性二次控制中的广义Riccati方程，证明其在正定矩阵锥上是局部收缩的，同时指出该流在其它Finsler度量下不再具有收缩性。