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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 $f : X \rightarrow \mathbb{R}$ with $f(x) \neq f(y)$ ? (where by "affine" I mean $\forall x',y' : f(m(x',y')) = \frac{f(x') + f(y')}{2})$ ?

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Is this definition of affine equivalent to the standard one (that a function is affine if its restriction to every geodesic is affine)? If yes, then the answer is given by "Spaces with many affine functions" by Hitzelberger-Lytchak. – Igor Belegradek Dec 19 '12 at 23:58
An example (the first example?) of a Hadamard space without this property is $\mathbb H^2$. – Will Sawin Dec 20 '12 at 0:51
I'd say that the tripod is even an easier example. – Misha Dec 20 '12 at 0:53
What's the tripod? – Will Sawin Dec 20 '12 at 3:26
Tripod is the metric graph obtained by gluing three segments at a common vertex. – Misha Dec 20 '12 at 4:02

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