Geodesic
Almost convex and Chebyshev sets
Matematičeskie zametki, Tome 8 (1970) no. 5, pp. 545-550 
L. I. Vlasov. Almost convex and Chebyshev sets. Matematičeskie zametki, Tome 8 (1970) no. 5, pp. 545-550. http://geodesic.mathdoc.fr/item/MZM_1970_8_5_a1/
@article{MZM_1970_8_5_a1,
     author = {L. I. Vlasov},
     title = {Almost convex and {Chebyshev} sets},
     journal = {Matemati\v{c}eskie zametki},
     pages = {545--550},
     year = {1970},
     volume = {8},
     number = {5},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1970_8_5_a1/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

Properties of normed spaces in which every almost convex set is convex are derived. In such spaces every Chebyshev set with a continuous metric projection is convex.