  The convex analytic method (generalized by Borkar) 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, under a near-monotone cost hypothesis, 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 stationary deterministic policies,
and these are under rather restrictive hypotheses. We provide mild conditions
under which an average cost optimal stochastic control problem admits optimal
solutions that are deterministic and stationary, building upon a study of
strategic measures by Feinberg. (iii) We establish conditions under which the
performance under stationary deterministic policies is dense in the set of
performance values under randomized stationary policies.
