Geodesic
Antiproximinal sets in spaces of continuous functions
Matematičeskie zametki, Tome 60 (1996) no. 5, pp. 643-657

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Closed convex bounded antiproximinal bodies are constructed in the infinite-dimensional spaces $C(Q)$, $C_0(T)$, $L_\infty(S,\Sigma,\mu)$ and $B(S)$, where $Q$ is a topological space and $T$ is a locally compact Hausdorff space. It is shown that there are no closed bounded antiproximinal sets in Banach spaces with the Radon–Nikodym property. 
@article{MZM_1996_60_5_a0,
     author = {V. S. Balaganskii},
     title = {Antiproximinal sets in spaces of continuous functions},
     journal = {Matemati\v{c}eskie zametki},
     pages = {643--657},
     publisher = {mathdoc},
     volume = {60},
     number = {5},
     year = {1996},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1996_60_5_a0/}
}
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V. S. Balaganskii. Antiproximinal sets in spaces of continuous functions. Matematičeskie zametki, Tome 60 (1996) no. 5, pp. 643-657. http://geodesic.mathdoc.fr/item/MZM_1996_60_5_a0/