Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 1 / 2 / a / Solution

A Hadamard space is a complete CAT(0) space. A useful equivalent definition, which makes the remaining assertions consequences rather than assumptions, is a complete metric space in which every pair $x,y$ has a midpoint $m$, with $d(x,m)=d(y,m)=d(x,y)/2$, satisfyingThis is the nonpositive-curvature midpoint inequality. Euclidean Hilbert spaces satisfy it with equality in the parallelogram formula. The comparison argument in (c) shows that this definition gives the usual CAT(0) triangle condition.