Geodesic
Defining a~metric in a~linear space by means of a~family of subsets
Matematičeskie zametki, Tome 19 (1976) no. 2, pp. 237-246

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Necessary and sufficient conditions are given on a family $\{A_r\}_{r>0}$ of subsets of a real linear space $X$ under which $\inf\{r>0:x\in A_r\}$ is a quasinorm [1] on X. It is shown that for any symmetric (about zero) closed set $A$ in a normed space $X$ containing the ball $\{x\in X:\|x\|\le1\}$ there exists a quasinorm $|\cdot|$ on $X$ such that $A=\{x\in X:\|x\|\le1\}$. Examples are constructed of linear metric spaces in which there exists a Chebyshev line which is not an approximately compact set. 
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     author = {A. I. Vasil'ev},
     title = {Defining a~metric in a~linear space by means of a~family of subsets},
     journal = {Matemati\v{c}eskie zametki},
     pages = {237--246},
     publisher = {mathdoc},
     volume = {19},
     number = {2},
     year = {1976},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1976_19_2_a8/}
}
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A. I. Vasil'ev. Defining a~metric in a~linear space by means of a~family of subsets. Matematičeskie zametki, Tome 19 (1976) no. 2, pp. 237-246. http://geodesic.mathdoc.fr/item/MZM_1976_19_2_a8/