Hilbert s Metric and Gromov Hyperbolicity
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1 Hilbert s Metric and Gromov Hyperbolicity Andrew Altman May 13,
2 1 HILBERT METRIC 2 1 Hilbert Metric The Hilbert metric is a distance function defined on a convex bounded subset of the n-dimensional Euclidean space R n. It was introduced as an example of a metric for which Euclidean straight lines are geodesics, and it generalizes Cayley s distance formula in the Cayley-Klein model of hyperbolic geometry. Two notable applications of the Hilbert metric are to Perron-Frobenius theory and to constructing Gromov hyperbolic spaces. This paper will focus on the latter of these applications. Definition. Let D be a bounded convex domain in R n, and for distinct x, y D, let a, b be the points at which the line xy intersects the boundary D of D where the order of the points is a, x, y, b. We define the Hilbert distance h as follows ( ) bx ay h(x, y) = log by ax where xy is the usual Euclidean distance. Furthermore by defining h(x, x) = 0 we obtain a metric on D. It is important to note t hat if one of the points x, y is on the boundary of D then h(x, y) = as is shown by a limiting process in We will denote the cross-ratio of four collinear points by [a, x, y, b] = bx ay by ax. Note that since the cross-ratio is preserved by linear fractional transformations, the Hilbert metric coincides with the Cayley-Klein metric on the open unit ball. 2 Gromov Hyperbolic The aim of this paper is to show, using the Hilbert metric, that a bounded convex domain which satisfies a certain intersecting chord property is Gromov hyperbolic (or δ-hyperbolic). By definition a Gromov hyperbolic space is a geodesic metric space in which all geodesic triangles are δ-thin, hence the notion of δ-hyperbolic.
3 3 INTERSECTING CHORDS PROPERTY 3 Definition. The Gromov product relative to z is (x, y) z = 1 (d(x, z) + d(y, z) d(x, y)). 2 Definition. A geodesic metric space (X, d) is said to be Gromov hyperbolic if for some δ 0 and for all w, x, y, z X (x, y) z min{(x, w) z, (w, y) z } δ We further expand the inequality which will be useful for later computations. Note we first multiplied both sides by 2 to take care of the 1 2 factor from the Gromov product. h(x, z) + h(z, y) h(x, y) min{h(x, z) + h(z, w) h(x, w), h(w, z) + h(z, y) h(w, y)} 2δ and rearranging the terms we obtain = min{h(x, z) h(x, w), h(z, y) h(w, y)} + h(w, z) 2δ h(x, y) + h(z, w) 2δ + h(x, z) + h(z, y) min{h(x, z) h(x, w), h(z, y) h(w, y)} = 2δ + h(x, z) + h(z, y) + max{h(x, w) h(x, z), h(w, y) h(z, y)} = 2δ + max{h(x, w) + h(z, y), h(w, y) + h(x, z)}. 3 Intersecting Chords Property Let c 1, c 2 be two intersecting chords in D; it is clear that these two interesting chords define a plane. Let l 1, l 1 and l 2, l 2 denote the respective lengths of the segments for which the chords c 1, c 2 have been divided. Then the domain D satisfies the intersecting chord property if for all δ > 0 there exists a constant C = C(D, δ) such that 1 C l 1l 1 C. l 2 l 2 We now consider the property of Menger curvature of any triple of points. Note that three points x, y, z in a plane not all collinear lie on a unique circle with radius R(x, y, z) = c 2 sin θ
4 4 THEOREM 4 where c is the length of the side of the triangle and θ the opposing angle in the triangle. Definition. Menger curvature of three points is defined to be the reciprocal of the radius R above and is denoted K(x, y, z). Proofs of the following statements can be found in [6]. Proposition. Let x 1, y 1 and x 2, y 2 be the respective endpoints of the chords c 1, c 2 defined above. Then l 1 l 1 l 2 l 2 = K(x 1, x 2, y 2 )K(y 1, x 2, y 2 ) K(x 2, x 1, y 1 )K(y 2, x 1, y 1 ) Corollary. Let D be a bounded convex domain in R n. Assume there is a constant C > 0 such that K(x, y, z) K(x, y, z ) C for any two distinct sets of triples in D all lying in the same 2-dimensional plane. Then D satisfies the intersecting chords property. 4 Theorem Theorem 1. Let h be the Hilbert metric and D be a bounded convex domain in R n satisfying the intersecting chords property. Then the metric space (D, h) is Gromov hyperbolic. Proof. Let D be as described such that the intersecting chord property holds for a constant C. Let y be a fixed reference point and x, z, w D any other three points. Then we obtain six chords which we will identify with by their cross-ratio, c 1 = [y, y, z, z ] c 2 = [y, y, w, w ] c 3 = [y, y, x, x ] c 4 = [w, w, x, x ] c 5 = [x, x, z, z ] c 6 = [w, w, z, z ]
5 4 THEOREM 5 Let H(x, z) = h(u, v)+h(w, y) h(u, w) h(v, y) then we must show that there is a constant independent of x, z, w such that min{h(x, z), H(z, w)} 2δ. First we can expand using the definition of h, ( ) xz zx wy yw xx ww zz yy H(x, z) = log xx zz ww yy xw wx zy yz ( ) xx xz yy yw zz zx ww wy = log xx xw yy yz zz zy zy yz By the intersecting chords property xx xx fractions), so that we have H(x, z) C + 2 log yw wy yz zy C xz xw ( ) xz yw zx wy. xw yz yz wx (and similarly for the other Furthermore y is a fixed point so is bounded as well giving ( ) xz H(x, z) C zx + 2 log. xw wx Hence we it suffices to check that { } xz zx min xw wx, zx xz zw wz is bounded. Without loss of generality we may assume zz xx by symmetry. Case 1: (xw xx or zw xx ) If xw zw then xz zx xz + zz )(zx + xx ) xw wx (xw) 2 (xw + wz + zz )(zw + wx + xx ) (xw) 2 (3xw)2 (xw) 2 = 9 Similarly the same argument applies for zw xw.
6 4 THEOREM 6 Case 2: (xw xx and zw xx ) Then since xx xz C(xx xw ), xz zx xw wx C xx zx xx wx C xx (xw + wz + xx ) wx xx 3C where zx = zx + xx xw + wz + xx since the sum of lengths of two sides of the triangle xwz are greater than or equal to the length of the third. Lastly we note that if D is the n-dimensional open unit ball B n = {(x 1,..., x n ) R n : n x 2 i < 1}. then by the Corollary (which is aided by intuition from the Proposition), the intersecting chords property holds for C = 1, and consequently the Theorem implies that the Cayley-Klein model of n-dimensional hyperbolic space is Gromov hyperbolic. i=1
7 REFERENCES 7 References [1] A. F. Beardon. The dynamics of contractions. Ergodic Theory Dynam. Systems, 17(6): , [2] Martin R. Bridson and André Haefliger. Metric spaces of non-positive curvature, volume 319 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, [3] Herbert Busemann. The geometry of geodesics. Academic Press Inc., New York, N. Y., [4] Herbert Busemann and Paul J. Kelly. Projective geometry and projective metrics. Academic Press Inc., New York, N. Y., [5] Ren Guo. A characterization of hyperbolic geometry among Hilbert geometry. J. Geom., 89(1-2):48 52, [6] Anders Karlsson and Guennadi A. Noskov. The Hilbert metric and Gromov hyperbolicity. Enseign. Math. (2), 48(1-2):73 89, 2002.
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