  Many iterative methods for solving optimization or feasibility problems have
been invented, and often convergence of the iterates to some solution is
proven. Under favourable conditions, one might have additional bounds on the
distance of the iterate to the solution leading thus to worst case estimates,
i.e., how fast the algorithm must converge.
  Exact convergence estimates are typically hard to come by. In this paper, we
consider the complementary problem of finding best case estimates, i.e., how
slow the algorithm has to converge, and we also study exact asymptotic rates of
convergence. Our investigation focuses on convex feasibility in the Euclidean
plane, where one set is the real axis while the other is the epigraph of a
convex function. This case study allows us to obtain various convergence rate
results. We focus on the popular method of alternating projections and the
Douglas-Rachford algorithm. These methods are connected to the proximal point
algorithm which is also discussed. Our findings suggest that the
Douglas-Rachford algorithm outperforms the method of alternating projections in
the absence of constraint qualifications. Various examples illustrate the
theory.
