  The method of alternating projections involves projecting an element of a
Hilbert space cyclically onto a collection of closed subspaces. It is known
that the resulting sequence always converges in norm and that one can obtain
estimates for the rate of convergence in terms of quantities describing the
geometric relationship between the subspaces in question, namely their pairwise
Friedrichs numbers. We consider the question of how best to order a given
collection of subspaces so as to obtain the best estimate on the rate of
convergence. We prove, by relating the ordering problem to a variant of the
famous Travelling Salesman Problem, that correctness of a natural form of the
Greedy Algorithm would imply that $\mathrm{P}=\mathrm{NP}$, before presenting a
simple example which shows that, contrary to a claim made in the influential
paper [Kayalar-Weinert, Math. Control Signals Systems, vol. 1(1), 1988], the
result of the Greedy Algorithm is not in general optimal. We go on to establish
sharp estimates on the degree to which the result of the Greedy Algorithm can
differ from the optimal result. Underlying all of these results is a
construction which shows that for any matrix whose entries satisfy certain
natural assumptions it is possible to construct a Hilbert space and a
collection of closed subspaces such that the pairwise Friedrichs numbers
between the subspaces are given precisely by the entries of that matrix.
