 Hadamard space

A Hadamard space is a complete CAT(0) space. Equivalently, every pair has a midpoint $m$ satisfying the midpoint inequality $d(z,m{)}^2\le \frac{1}{2}d(z,x{)}^2+\frac{1}{2}d(z,y{)}^2-\frac{1}{4}d(x,y{)}^2$ for every $z$. This inequality makes the midpoint unique. Iterating midpoints and using completeness constructs the unique geodesic joining the pair. The squared-distance inequality extends to every point of that geodesic by dyadic approximation and implies Euclidean triangle comparison and geodesic convexity of closed balls.

A **Hadamard space** is a specific type of metric space that generalizes the concept of non-positive curvature. More formally, a Hadamard space is a complete metric space where any two points can be connected by a geodesic, and all triangles in the space are "thin" in a sense that closely resembles the behavior of triangles in hyperbolic geometry.