Uniform Convexity of Geodesic Ptolemy Spaces

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