Geodesic
The conditional Chebyshev center of a compact set of continuous functions
Matematičeskie zametki, Tome 14 (1973) no. 4, pp. 469-478 Cet article a éte moissonné depuis la source Math-Net.Ru

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We establish characteristic properties of a subspace $L$ of finite codimension of the space $C(T)$ that has a Chebyshev center and a Chebyshev net for every compact set from $C(T)$. We show that these properties are the same as the conditions for the existence in $L$ of an element of best approximation for every element from $C(T)$. 
@article{MZM_1973_14_4_a1,
     author = {A. L. Garkavi},
     title = {The conditional {Chebyshev} center of a~compact set of continuous functions},
     journal = {Matemati\v{c}eskie zametki},
     pages = {469--478},
     year = {1973},
     volume = {14},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1973_14_4_a1/}
}
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A. L. Garkavi. The conditional Chebyshev center of a compact set of continuous functions. Matematičeskie zametki, Tome 14 (1973) no. 4, pp. 469-478. http://geodesic.mathdoc.fr/item/MZM_1973_14_4_a1/