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Journal of Convex Analysis 20 (2013), No. 3, 689--700
Copyright Heldermann Verlag 2013

Uniform Convexity of Geodesic Ptolemy Spaces

Rafa Espínola
Dep. de Análisis Matemático - IMUS, Universidad de Sevilla, Apdo. de Correos 1160, 41080 Sevilla, Spain

Adriana Nicolae
Dept. of Mathematics, Babes-Bolyai University, Kogalniceanu 1, 400084 Cluj-Napoca, Romania

It has been recently proved that balls in a proper geodesic Ptolemy space are strictly convex. In this work we further study Ptolemy spaces in this direction. In particular we prove the uniform convexity of geodesic Ptolemy spaces when assuming a uniform continuity condition on a midpoint map and use this property to deduce different geometrical and convexity related facts about such spaces. We also show that a nice modulus of uniform convexity can be found in such a setting.

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