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.
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A Hadamard space is a complete CAT(0) space. Equivalently, every pair has a midpoint satisfying the midpoint inequality for every . 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.