A remark on the sum of proximinal subspaces (Q1263772)

\textit{E. W. Cheney} and \textit{D. E. Wulbert} [Math. Scand. 24, 113-140 (1969; Zbl 0186.187)] asked the following question: if U and V are proximinal subspaces of a Banach space X and if \(U+V\) is closed, does it follow that \(U+V\) is proximinal? \textit{M. Feder} [J. Approximation Theory 49, 144-148 (1987; Zbl 0623.41036)] show that the answer to this question is negative. In the present paper the strengthening of this result is proved. In particular, is valid: Theorem 3. Let U be a Banach space. Then U is reflexive if and only if for any Banach space \(X\supseteq U\), V is proximinal in X and \(U+V\) is closed implies \(U+V\) is proximinal in X.