Characterisations of Chebyshev sets in \(c_{0}\) (Q1885447)

A subset \(M\) of a normal linear space \(X\) is called a Chebyshev set if there exists a unique best approximation in \(M\) for each \(x\in X\). A characterization of Chebyshev sets which is known for \(l^\infty (n)\) is extended to \(c_0\). A central role play the \textit{coordinate affine subspaces}, i.e. the subspaces of functions which vanish at \(k\) points \((k\in \mathbb N)\) -- and their translations. Two conditions characterize Chebyshev sets. The first one says that the intersections with coordinate affine subspaces are connected. The second one says that locally supporting planes of that kind have an analogous property also globally.