  Waist inequality is a fundamental inequality in geometry and topology. We
apply it to the study of entropy and mean dimension of dynamical systems. We
consider equivariant continuous maps between dynamical systems and assume that
the mean dimension of the domain is larger than the mean dimension of the
target. We exhibit several situations for which the maps necessarily have
positive conditional metric mean dimension. This study has interesting
consequences to the theory of topological conditional entropy. In particular it
sheds new light on a celebrated result of Lindenstrauss and Weiss about minimal
dynamical systems non-embeddable in the shift on the Hilbert cube.
