Geodesic
Convex antiproximal sets in spaces $c_0$ and $c$
Matematičeskie zametki, Tome 17 (1975) no. 3, pp. 449-457.

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In the note we prove that in a Banach space c there exists a closed bounded symmetric convex division ring $V_1$ such that for any $x\in c\setminus V_1$, $P_{V_1}(x)=\emptyset$ where $P_{V_1}$ is the metric projection onto $V_1$. 
@article{MZM_1975_17_3_a11,
     author = {S. Kobzash},
     title = {Convex antiproximal sets in spaces $c_0$ and $c$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {449--457},
     publisher = {mathdoc},
     volume = {17},
     number = {3},
     year = {1975},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1975_17_3_a11/}
}
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S. Kobzash. Convex antiproximal sets in spaces $c_0$ and $c$. Matematičeskie zametki, Tome 17 (1975) no. 3, pp. 449-457. http://geodesic.mathdoc.fr/item/MZM_1975_17_3_a11/