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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:
 [1] S. Alexander and R. Bishop, A cone splitting theorem for Alexandrov spaces, Pacific J. Math. 218 (2005). · Zbl 1125.53023 [2] N. Innami, Splitting theorems of Riemannian manifolds, Compositio Math. 47 (1982), 237-247. · Zbl 0514.53040 [3] A. Lytchak and V. Schroeder, Affine functions on $$(\textit{CAT}(\kappa))$$ spaces, Math. Z. 255 (2007), 231-244. · Zbl 1197.53044 [4] A. Lytchak, Differentiation in metric spaces, Algebra i Analiz 16 (2004), no. 6, 128 – 161; English transl., St. Petersburg Math. J. 16 (2005), no. 6, 1017 – 1041. [5] Yukihiro Mashiko, A splitting theorem for Alexandrov spaces, Pacific J. Math. 204 (2002), no. 2, 445 – 458. · Zbl 1058.53036 [6] V. Matveev, Hyperbolic manifolds are geodesically rigid, Invent. math 151 (2003), 579-609. · Zbl 1039.53046 [7] Shin-ichi Ohta, Totally geodesic maps into metric spaces, Math. Z. 244 (2003), no. 1, 47 – 65. · Zbl 1046.53047
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