Geodesic
A continuity criterion for metric projections in Banach spaces
Matematičeskie zametki, Tome 10 (1971) no. 4, pp. 459-468.

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The paper provides the complete characteristics of the class of reflexive strictly convex spaces in which the metric projections on each convex closed set are continuous. 
@article{MZM_1971_10_4_a10,
     author = {E. V. Oshman},
     title = {A continuity criterion for metric projections in {Banach} spaces},
     journal = {Matemati\v{c}eskie zametki},
     pages = {459--468},
     publisher = {mathdoc},
     volume = {10},
     number = {4},
     year = {1971},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1971_10_4_a10/}
}
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E. V. Oshman. A continuity criterion for metric projections in Banach spaces. Matematičeskie zametki, Tome 10 (1971) no. 4, pp. 459-468. http://geodesic.mathdoc.fr/item/MZM_1971_10_4_a10/