## On Hilbert’s metric for simplices.(English)Zbl 0832.52002

Niblo, Graham A. (ed.) et al., Geometric group theory. Volume 1. Proceedings of the symposium held at the Sussex University, Brighton (UK), July 14-19, 1991. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 181, 97-119 (1993).
The author investigates the isometries of the canonical Hilbert metric defined on any bounded convex open subset $$C$$ of a finite dimensional real vector space $$V$$. In case $$C$$ is an open 2-simplex $$X$$, it is shown that the resulting space is isometric to $$\mathbb{R}^2$$ with a norm such that the unit ball is a regular hexagon and that the central symmetry in this plane corresponds to the quadratic transformation associated to $$S$$. Finally, Hilbert’s metric for symmetric spaces is discussed briefly and some open problems are stated.
For the entire collection see [Zbl 0777.00044].