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Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.

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Existence of non-trivial affine functions on Hadamard spaces ?

Let $X$ be a Hadamard space. Any two points $x$ and $y$ of $X$ have a unique midpoint $m = m(x,y)$. Given $x$ and $y$ any two points of $X$, is it always possible to find an affine function $f : X \r …