Geodesic
The Geometric Structure of Chebyshev Sets in $\ell^\infty(n)$
Funkcionalʹnyj analiz i ego priloženiâ, Tome 39 (2005) no. 1, pp. 1-10

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A subset $M$ of a normed linear space $X$ is called a Chebyshev set if each $x\in X$ has a unique nearest point in $M$. We characterize Chebyshev sets in $\ell^\infty(n)$ in geometric terms and study the approximative properties of sections of Chebyshev sets, suns, and strict suns in $\ell^\infty(n)$ by coordinate subspaces.
Keywords: Chebyshev set, sun, strict sun, best approximation. 
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     title = {The {Geometric} {Structure} of {Chebyshev} {Sets} in $\ell^\infty(n)$},
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A. R. Alimov. The Geometric Structure of Chebyshev Sets in $\ell^\infty(n)$. Funkcionalʹnyj analiz i ego priloženiâ, Tome 39 (2005) no. 1, pp. 1-10. http://geodesic.mathdoc.fr/item/FAA_2005_39_1_a0/