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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
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References:
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