  This paper contributes to the mean dimension theory of dynamical systems. We
introduce a new concept called mean dimension with potential and develop a
variational principle for it. This is a mean dimension analogue of the theory
of topological pressure. We consider a minimax problem for the sum of rate
distortion dimension and the integral of a potential function. We prove that
the minimax value is equal to the mean dimension with potential for a dynamical
system having the marker property. The basic idea of the proof is a
dynamicalization of geometric measure theory.
