On nearest and farthest points (Q1272286)

Let \(X\) be a real Banach space and let \(S\) be its unit sphere. We write \(X\in (RN),\) (\(X\in (CLUR)\) respectively), if \(X\) has the Radon-Nikodým property, (if the conditions \(x, x_n\in S\), \(f\in S^*\), \(f(x)=1\), \(f(x_n)\rightarrow 1\) imply that \(x_n\) has a convergent subsequence, respectively). A nonempty subset \(A\subsetneqq X\) is said to be antiproximinal (antiremotal) if for any point \(x\in X\setminus A, (x\in X)\) the set \(A\) does not contain a nearest (farthest) point. Main results: a) If \(X\notin (RN)\) then \(X\) contains a bounded closed centrally symmetric antiremotal set. b) If \(X\in (CLUR)\setminus (RN),\) then \(X\) contains a closed antiproximinal set whose complement is a convex bounded centrally symmetric body. For example, the author proves that \(A =\{ x\in L_1[0,1]: \int_0^1(1+t)| x(t)| dt\leq 1\}\) is antiremotal and \(M=\overline{L_1[0,1]\setminus A}\) is antiproximinal.