Injective Metric Space - Hyperconvexity

Hyperconvexity

A metric space X is said to be hyperconvex if it is convex and its closed balls have the binary Helly property. That is,

1. any two points x and y can be connected by the isometric image of a line segment of length equal to the distance between the points (i.e. X is a path space), and
2. if F is any family of closed balls

such that each pair of balls in F meet, then there exists a point x common to all the balls in F.

Equivalently, if a set of points p_i and radii r_i > 0 satisfies r_i + r_j ≥ d(p_i,p_j) for each i and j, then there is a point q of the metric space that is within distance r_i of each p_i.