  We study variational principles for metric mean dimension. First we prove
that in the variational principle of Lindenstrauss and Tsukamoto it suffices to
take supremum over ergodic measures. Second we derive a variational principle
for metric mean dimension involving growth rates of measure-theoretic entropy
of partitions decreasing in diameter which holds in full generality and in
particular does not necessitate the assumption of tame growth of covering
numbers. The expressions involved are a dynamical version of Renyi information
dimension. Third we derive a new expression for Geiger-Koch information
dimension rate for ergodic shift-invariant measures. Finally we develop a lower
bound for metric mean dimension in terms of Brin-Katok local entropy.
