Quasi-orthogonality of the best approximant sets (Q2499614)

A concept of quasi-orthogonality in normed linear spaces is introduced and applied to obtain some results about best approximations in normed linear spaces which were obtained earlier in Hilbert spaces. Let \(\widehat M=\{x\in X\mid \text{dist} (x,M)=\| x\| \}\). Suppose \(X\) is a rotund (strictly convex) space and \(M\) is a proximinal subset of \(X\). If \(\widehat M\) is convex, then \(M\) is a Chebyshev set.