The Hilbert metric and Gromov hyperbolicity. (Q1419569)

Given a convex domain \(D\) in the Euclidean space, for any pair of points \(x\) and \(y\) in \(D\) let us denote by \(x^\prime\) and \(y^\prime\) the intersections of the line through \(x\) and \(y\) with the boundary of \(D\) closest to \(x\) and \(y\). The logarithm of the crossratio of these four points defines the Hilbert metric on \(D\): \(h(x,y) = \log \frac{| yx^\prime| | xy^\prime| }{| xx^\prime| | yy^\prime| }\). Such a metric in the unit disk gives a model of hyperbolic space. It is said that a domain has the intersecting chords property if there is a positive constant \(M\) such that for any intersecting chords \(c_1\) and \(c_2\) in \(D\) we have \(M^{-1} \leq \frac{l_1 l_1^\prime}{l_2 l_2^\prime} \leq M\) where the intersection point divides the chord \(c_j\) into segments of the lengths \(l_j\) and \(l_j^\prime\), \(j=1,2\). The authors prove that if a convex domain \(D\) has the intersections chords property then this domain endowed with the Hilbert metric is Gromov hyperbolic.