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Spaces with many affine functions. (English) Zbl 1128.53021
Let $$X$$ be a geodesically connected metric space. A function $$f:X\rightarrow \mathbb{R}$$ is said to be affine if $$f\left( \gamma\left( s\right) \right) =as+b$$ for every normal geodesic $$\gamma$$ in $$X$$, where numbers $$a$$ and $$b$$ may depend on $$\gamma$$; $$X$$ is separated by affine functions if, for every pair of distinct points $$x,y\in X$$, there is an affine function $$f$$ on $$X$$ such that $$f\left( x\right) \neq f\left( y\right)$$. The main result states that points of $$X$$ are separated by affine functions if and only if $$X$$ is isometric to a convex subset of a normed space with a strictly convex norm.

##### MSC:
 53C20 Global Riemannian geometry, including pinching 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions 53C22 Geodesics in global differential geometry
##### Keywords:
affine functions; norms; geodesic mappings
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##### References:
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