A note on coapproximation (Q1184917)

Let \(S\) be a set of a normed space \(N\). An element \(s_ 0\in S\) is said to be a coapproximation of \(x\in N\) if \(\| s-s_ 0\|\leq\| x- s\|\) for all \(s\in S\). The authors prove the following results. If \(S\) is a zero hyperplane of \(N\) and \(u\) is single valued, where \(u(x)=\{s_ 0\in S:\;\| s-s_ 0\|\leq\| x-s\|\;\forall s\in S\}\) for \(x\in N\), then \(u\) is linear and \(\| u\|=1\); If \(S\) is a convex subset, then \(u(x)\) is a convex subset of \(S\); If \(N\) is a smooth normed space and \(S\) is a subspace of \(N\), then \(u(x)\) is single valued whenever \(u(x)\neq\emptyset\).