Vol. 25, No. 3, 1968

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Curvature in Hilbert geometries. II

Paul Joseph Kelly and Ernst Gabor Straus

Vol. 25 (1968), No. 3, 549–552
Abstract

The interior of a closed convex curve C in the Euclidean plane can be given a Hilbert metric, which is preserved by projective mappings. Let p, q be points interior to C and let u, v be the points of intersection of the line pq with C. The Hilbert distance h(p,q) is defined by

            d(u,p)d(v,q)-
h(p,q) = |log d(v,p)d(u,q)|,

where d(x,y) denotes Euclidean distance. If C contains at most one line segment then h(p,q) is a proper metric and the metric lines are the open chords of C carried by the Euclidean lines. Following Busemann [1, p. 237], we define the (qualitative) curvature at a point p as positive or negative if there exists a neighborhood U of p such that for every x,y U we have

   ----                        ----
2h(x,y) ≧ h(x,x) respectively  2h(x,y) ≦ h(x,y),

where x y are the Hilbert midpoints of p and x and of p and y respectively.

In an earlier paper [2] we proved that any point p at which the sign of the curvature is determined is a projective center of C; that is, there exists a projective transformation which maps p into an affine center of the image of C. We also stated the conjecture that a Hilbert geometry has no point of positive curvature. It is the purpose of this paper to prove that conjecture.

Mathematical Subject Classification
Primary: 52.50
Milestones
Received: 17 September 1967
Published: 1 June 1968
Authors
Paul Joseph Kelly
Ernst Gabor Straus