The geometric structure of Chebyshev sets in \(\ell^\infty(n)\) (Q2571488)

Let \(M\) (resp.\(H\)) be a set (resp. subspace) in \(R^n\). Some approximative properties of the set \(M\cap N \) in the subspace \(H\), where the norm on \(H\) is induced from \(l^\infty (n) \), is studied. The results for coordinate and noncoordinate subspaces \(H\) are completely different. There are two main results in the paper. Theorem 1 describes the approximative properties of intersections of Chebyshev sets, suns, and strict suns in \(l^\infty (n) \) with coordinate subspaces. Theorem 2 characterizes Chebyshev sets in \(l^\infty (n) \) in geometric terms.