Hadamard space

A Hadamard space is a complete CAT(0) space. Equivalently, it is a complete metric space in which every pair $x,y$ has a midpoint $m$ satisfyingfor every $z$. A midpoint means $d(x,m)=d(m,y)=d(x,y)/2$. Substituting a second midpoint for $z$ forces their distance to be zero, so the midpoint is unique.

Iterating midpoints defines the joining geodesic first at dyadic parameters. Completeness then supplies its points at every real parameter. The midpoint inequality extends by this approximation to the Hadamard squared-distance inequality, and hence to Euclidean triangle comparison. In particular, closed balls have geodesic convexity.

The completeness condition belongs in both formulations. The rational numbers with the usual distance have all rational midpoints and satisfy the displayed inequality, but are incomplete. A full real-parameter geodesic from zero to one would require irrational points. Thus the midpoint inequality alone does not make that space a Hadamard space.

For the constant-speed geodesic $\gamma$ from $x$ to $y$ in a Hadamard space, the displayed inequality follows by repeatedly applying the midpoint inequality and passing from dyadic parameters to all $t\in [0,1]$. Applying it on two sides of a triangle gives $d(x_s,y_t{)}^2\le s^2{A}^2+t^2{B}^2-st(A^2+B^2-C^2)$, where $A=d(a,x)$, $B=d(a,y)$ and $C=d(x,y)$. The right side is the squared comparison distance by the Euclidean cosine rule.