The transitive property of parallel lines is a characteristic property of real strictly convex Banach spaces
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- by J. E. Valentine PDF
- Proc. Amer. Math. Soc. 95 (1985), 604-606 Request permission
Abstract:
In a recent paper Freese and Murphy said a complete, convex, externally convex metric space has the vertical angle property provided for each four of its distinct points $p$, $q$, $r$, $s$, if $m$ is a midpoint of $p$ and $q$ and of $r$ and $s$, then $pr = qs$. In this paper we say a line $L$ is parallel to a line $N$ in such a space provided $L$ and $N$ contain points $p$, $r$, and $q$, $s$, respectively, such that the segments $S\left ( {p,q} \right )$ and $S\left ( {r,s} \right )$ have a common midpoint $m$. We further assume that if line $L$ is parallel to line $N$ and line $N$ is parallel to line $R$, then $L$ is parallel to $R$. The main result of this paper is that such a space is a real strictly convex Banach space. Since real strictly convex Banach spaces have all of the above properties, the characterization is then complete.References
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Additional Information
- © Copyright 1985 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 95 (1985), 604-606
- MSC: Primary 51K05; Secondary 46B20
- DOI: https://doi.org/10.1090/S0002-9939-1985-0810171-2
- MathSciNet review: 810171