  We prove a variational principle for the metric mean dimension analog to the
one in [LT]. Instead of using the rate distortion function we use the function
$h_\mu(\epsilon,T,\delta)$ that is closely related to the entropy $h_\mu(T)$ of
$\mu$. Our formulation has the advantage of being, in the authors opinion, more
natural when doing computations. As a corollary we obtain a proof of the
standard variational principle. We also obtain some relations between the rate
distortion function with our function $\tilde{h}_\mu(\epsilon,T,\delta)$, a
modification of $h_\mu(\epsilon,T,\delta)$ when replacing the dynamical metrics
with the average dynamical metrics. Using our methods we also reprove the main
result in [LT]. We will explain how to construct homeomorphisms on closed
manifolds with maximal metric mean dimension. We end this paper with some
questions that naturally arise from this work.
