Hadamard space

In geometry, an Hadamard space, named after Jacques Hadamard, is a non-linear generalization of a Hilbert space. In the literature they are also equivalently defined as complete CAT(0) spaces.

A Hadamard space is defined to be a nonempty complete metric space such that, given any points x, y, there exists a point m such that for every point z,

The point m is then the midpoint of x and y: ${\displaystyle d(x,m)=d(y,m)=d(x,y)/2}$.

In a Hilbert space, the above inequality is equality (with ${\displaystyle m=(x+y)/2}$), and in general an Hadamard space is said to be flat if the above inequality is equality. A flat Hadamard space is isomorphic to a closed convex subset of a Hilbert space. In particular, a normed space is an Hadamard space if and only if it is a Hilbert space.

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