Projective distance and g-measures

Abstract

In 1895 D. Hilbert introduced, in an early paper on foundations on geometry, the projective metric on the positive cone in Rn [15]. The Hilbert projective distance dp (de ned on pairs of vectors with positive elements in corresponding positions) is a pseudo-metric, dp(x; y) = 0 if and only if x = y for some scalar > 0. In a rst glance to this metric (more details in Section 1.2), we can see that its de nition is rather complicated, so, the rst obvious question consists of asking if there exists a \less complicated" way to de ne it, and of course, guarantee that with this de nition we still have the contractive properties that characterize it. The answer is given by Kohlberg and Pratt [20] who proved that Hilbert's dp is essentially the only metric de ned on the positive cone of Rn which makes positive linear transformations contractive mappings with respect to this distance. Any projective metric, say ~ d, de ned on this cone such that every positive linear transformation is a contraction with respect to ~ d is equivalent to dp in the following sense: ~ d(x; y) = f(dp(x; y)) for a continuous, positive and strictly increasing function f and for all positive vectors x; y. Bushell [4] gave elementary derivations of the principal properties of Hilbert's metric in a general (Banach space) setting, namely the triangle inequality and completeness criteria in positive cones contained in di erent metric spaces. The considered spaces are the positive cone in Rn, the set of continuous positive functions de ned on the unitary interval, the cone of real positive semide nite symmetric matrices and Banach lattices. Numerous applications have arisen from this theory, namely, contributions to the theory of non-negative matrices, positive integral operators, positive-de nite symmetric matrices and the study of solutions to systems of ordinary di erential equations.