Geodesic
Example of an antiproximinal, but not a 2-antiproximinal convex closed bounded body
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 5 (2016), pp. 49-52

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An example of an antiproximinal but not 2-antiproximinal convex closed bounded body is constructed in the space $\bf{c}_0$ endowed with Day's norm. 
@article{VMUMM_2016_5_a8,
     author = {B. B. Bednov},
     title = {Example of an antiproximinal, but not a 2-antiproximinal convex closed bounded body},
     journal = {Vestnik Moskovskogo universiteta. Matematika, mehanika},
     pages = {49--52},
     publisher = {mathdoc},
     number = {5},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMUMM_2016_5_a8/}
}
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B. B. Bednov. Example of an antiproximinal, but not a 2-antiproximinal convex closed bounded body. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 5 (2016), pp. 49-52. http://geodesic.mathdoc.fr/item/VMUMM_2016_5_a8/