  Firstly, we answer the problem 1 asked by Gutman and $\rm \acute{\ S}$piewak
in \cite{gs20}, then we establish a double variational principle for mean
dimension in terms of R$\bar{e}$nyi information dimension and show the order of
$\sup$ and $\limsup$ (or $\liminf$) of the variational principle for the metric
mean dimension in terms of R$\bar{e}$nyi information dimension obtained by
Gutman and $\rm \acute{\ S}$piewak can be changed under the marker property.
Finally, we attempt to introduce the notion of maximal metric mean dimension
measure, which is an analogue of the concept called classical maximal entropy
measure related to the topological entropy.
