Geodesic
Operator of best approximation on finite-dimensional subspaces
Matematičeskie zametki, Tome 16 (1974) no. 3, pp. 501-511

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The modulus of continuity of the operator of best approximation on a subspace tends towards zero uniformly on the class of all subspaces of an n-dimensional space only if the unit ball of the space contains no extremal subsets of dimensionality $k$ ($0$). 
@article{MZM_1974_16_3_a18,
     author = {V. I. Berdyshev},
     title = {Operator of best approximation on finite-dimensional subspaces},
     journal = {Matemati\v{c}eskie zametki},
     pages = {501--511},
     publisher = {mathdoc},
     volume = {16},
     number = {3},
     year = {1974},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1974_16_3_a18/}
}
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V. I. Berdyshev. Operator of best approximation on finite-dimensional subspaces. Matematičeskie zametki, Tome 16 (1974) no. 3, pp. 501-511. http://geodesic.mathdoc.fr/item/MZM_1974_16_3_a18/