Existence of non-trivial affine functions on Hadamard spaces ? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T06:06:13Z http://mathoverflow.net/feeds/question/116828 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/116828/existence-of-non-trivial-affine-functions-on-hadamard-spaces Existence of non-trivial affine functions on Hadamard spaces ? km 2012-12-19T22:54:41Z 2012-12-19T22:54:41Z <p>Let $X$ be a Hadamard space. Any two points $x$ and $y$ of $X$ have a unique midpoint $m = m(x,y)$.</p> <p>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 $\forall x',y' : f(m(x',y')) = \frac{f(x') + f(y')}{2})$ ? </p>