  In this article, we introduce a notion of relative mean metric dimension with
potential for a factor map $\pi: (X,d, T)\to (Y, S)$ between two topological
dynamical systems. To link it with ergodic theory, we establish four
variational principles in terms of metric entropy of partitions, Shapira's
entropy, Katok's entropy and Brin-Katok local entropy respectively. Some
results on local entropy with respect to a fixed open cover are obtained in the
relative case. We also answer an open question raised by Shi \cite{Shi}
partially for a very well-partitionable compact metric space, and in general we
obtain a variational inequality involving box dimension of the space.
Corresponding inner variational principles given an invariant measure of
$(Y,S)$ are also investigated.
