An alternating projection that does not converge in norm (Q1881089)

Let \(C_1\) and \(C_2\) be two intersecting closed convex sets in a Hilbert space. Let \(P_1\) and \(P_2\) denote the corresponding projection operators. In 1933, von Neumann proved that the iterates produced by the sequence of alternating projections defined as \({y_n=(P_1 P_2)^n y_0}\) converge in norm to \({P_{C_1\cap C_2}(y_0)}\) when \(C_1\) and \(C_2\) are closed subspaces. \textit{L. M.~Bregman} [Sov. Math., Dokl. 6, 688--692 (1965; Zbl 0142.16804)] showed that the iterates converge weakly to a point in \({C_1\cap C_2}\) for any pair of closed convex sets. In the paper under review, the author shows that alternating projections not always converge in the norm by constructing an explicit counterexample.