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 ℝ2 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 ℓx,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.