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 piolemaicinequality
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.
