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

de la Harpe, Pierre

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].

Reviewer: H.Azad (Islamabad)

MSC:

52A21

Convexity and finite-dimensional Banach spaces (including special norms, zonoids, etc.) (aspects of convex geometry)

Keywords:

isometries; central symmetry; quadratic transformation; Hilbert’s metric; symmetric spaces