The Douglas-Rachford algorithm for the case of the sphere and the line

In this paper, we solve a conjecture proposed by Borwein and Sims (Fixed-point algorithms for inverse problems in science and engineering, Springer optimization and its applications, 2011) in a Hilbert space setting. For the simple non-convex example of the sphere and the line, the sequence of Douglas-Rachford iterates converges in norm to a point of the intersection except when the initial value belongs to the hyperplane of symmetry.