On Hibert's Metric for Simplices

Abstract

Abstract. For any bounded convex open subset C of a finite dimensional real vector space, we review the canonical Hilbert metric defined on C and we investigate the corresponding group of isometries. In case C is an open 2-simplex S , we show that the resulting space is isometric to 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, we discuss briefly Hilbert's metric for symmetric spaces and we state some open problems. Generalities on Hilbert metrics The first proposition below comes from a letter of D. Hilbert to F. Klein [Hil]. It is discussed in several other places, such as sections 28, 29 and 50 of [BuK], and chapter 18 of [Bui], and [Bea]. There are also nice applications of Hilbert metrics to the classical Perron-Frobenius Theorem [Sae], [KoP] and to various generalizations in functional analysis [Bir], [Bus]. Let V be a real affine space, assumed here to be finite dimensional (except in Remark 3.3), and let C be a non empty bounded convex open subset of V . We want to define a metric on C which, in the special case where C is the open unit disc of the complex plane, gives the projective model of the hyperbolic plane (sometimes called the “Klein model”). Let x,y ∈ C . If x = y , one sets obviously d ( x,y ) = 0. Otherwise, the well defined affine line l x,y ⊂ V containing x and y cuts the boundary of C in two points, say u on the side of x and v on the side of y ; see Figure 1.