  We focus on the convergence analysis of averaged relaxations of cutters,
specifically for variants that---depending upon how parameters are
chosen---resemble \emph{alternating projections}, the \emph{Douglas--Rachford
method}, \emph{relaxed reflect-reflect}, or the \emph{Peaceman--Rachford}
method. Such methods are frequently used to solve convex feasibility problems.
The standard convergence analyses of projection algorithms are based on the
\emph{firm nonexpansivity} property of the relevant operators. However if the
projections onto the constraint sets are replaced by cutters (projections onto
separating hyperplanes), the firm nonexpansivity is lost. We provide a proof of
convergence for a family of related averaged relaxed cutter methods under
reasonable assumptions, relying on a simple geometric argument. This allows us
to clarify fine details related to the allowable choice of the relaxation
parameters, highlighting the distinction between the exact (firmly
nonexpansive) and approximate (strongly quasi-nonexpansive) settings. We
provide illustrative examples and discuss practical implementations of the
method.
