Vol. 21, No. 2, 1967

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The ptolemaic inequality in Hilbert geometries

David Clifford Kay

Vol. 21 (1967), No. 2, 293–301

Let M be a metric space, and if x and y are points in M, let xy denote the metric. The space M and its metric are called ptolemaic if for each quadruple of points xi(i = 1,2,3,4) the piolemaic inequality

x1x2 ⋅x3x4 + x1x3 ⋅x2x4 ≧ x1x4 ⋅x2x3

holds. If the inequality holds only in some neighborhood of each point the space and its metric are said to be locally ptolemaic. Euclidean space is known to be ptolemaic and therefore, locally ptolemaic. We are interested here in certain non-euclidean spaces which may possibly be locally ptolemaic. The author has proved in his thesis (Michigan State University Doctoral Dissertation, 1963) that a Riemannian geometry is locally ptolemaic if and only if it has nonpositive curvature, and that a Finsler space which is locally ptolemaic is Riemannian. The main result established here extends the theorem regarding Finsler spaces to include Hilbert geometries as well: A Hilbert geometry is locally ptolemaic if and only if it is hyperbolic.

Mathematical Subject Classification
Primary: 52.30
Received: 27 April 1965
Published: 1 May 1967
David Clifford Kay