  The convex analytic method has proved to be a very versatile method for the
study of infinite horizon average cost optimal stochastic control problems. In
this paper, we revisit the convex analytic method and make three primary
contributions: (i) We present an existence result for controlled Markov models
that lack weak continuity of the transition kernel but are strongly continuous
in the action variable for every fixed state variable. (ii) For average cost
stochastic control problems in standard Borel spaces, while existing results
establish the optimality of stationary (possibly randomized) policies, few
results are available on the optimality of deterministic policies. We review
existing results and present further conditions under which an average cost
optimal stochastic control problem admits optimal solutions that are
deterministic stationary. (iii) We establish conditions under which the
performance under stationary deterministic (and also quantized) policies is
dense in the set of performance values under randomized stationary policies.
