  Topological data analysis asks when balls in a metric space $(X,d)$
intersect. Geometric data analysis asks how much balls have to be enlarged to
intersect. We connect this principle to the traditional core geometric concept
of curvature. This enables us, on one hand, to reconceptualize curvature and
link it to the geometric notion of hyperconvexity. On the other hand, we can
then also understand methods of topological data analysis from a geometric
perspective.
