= Hadamard space
{c}
A <Hadamard space> is a complete <CAT(0) space>. Equivalently, every pair has a midpoint $m$ satisfying the midpoint inequality $d(z,m)^2\leq\tfrac12d(z,x)^2+\tfrac12d(z,y)^2-\tfrac14d(x,y)^2$ 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.
Back to article page
