Non-positively curved metric spaces

Having an upper or lower bound for the sectional curvature is equivalent to some metric properties which are geometrically meaningful even in geodesic length spaces. This led to the synthetic theory of curvature bounds in geodesic length spaces or more precisely the Alexandrov and Busemann definitions of curvature bounds. These extensions to metric geometry still require that any two points can be connected by a shortest geodesic.

In our current project, on which this talk is based, we introduce a notion of curvature inequalities that works with intersection patterns of distance balls and therefore is meaningful even for discrete metric spaces.

Such intersection patterns have already been investigated from different perspectives in persistent homology method in topological data analysis which (using Čech homology groups) records how such intersection patterns change when the radii of those distance balls increase.

Unlike previous developments (where the extreme space is Euclidean plane), extremes of our classifications are tripod spaces including hyperbconvex spaces (which have trivial Čech homology groups). So we list some properties of hyperconvex spaces and some topological results in tripod spaces as well.