Geodesic
Antiproximinal convex bounded sets in the space $c_0(\Gamma)$ equipped with the day norm
Matematičeskie zametki, Tome 79 (2006) no. 3, pp. 323-338

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We construct a convex smooth antiproximinal set in any infinite-dimensional space $c_0(\Gamma)$ equipped with the Day norm; moreover, the distance function to the set is Gâteaux differentiable at each point of the complement. 
@article{MZM_2006_79_3_a0,
     author = {V. S. Balaganskii},
     title = {Antiproximinal convex bounded sets in the space $c_0(\Gamma)$ equipped with the day norm},
     journal = {Matemati\v{c}eskie zametki},
     pages = {323--338},
     publisher = {mathdoc},
     volume = {79},
     number = {3},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2006_79_3_a0/}
}
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V. S. Balaganskii. Antiproximinal convex bounded sets in the space $c_0(\Gamma)$ equipped with the day norm. Matematičeskie zametki, Tome 79 (2006) no. 3, pp. 323-338. http://geodesic.mathdoc.fr/item/MZM_2006_79_3_a0/