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
π
:
(
X
,
T
)
→
(
Y
,
S
)
\pi : (X, T) \to (Y, S)
between dynamical systems and assume that the mean dimension of the domain
(
X
,
T
)
(X, T)
is larger than the mean dimension of the target
(
Y
,
S
)
(Y, S)
. We exhibit several situations for which the maps
π
\pi
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 [Israel J. Math. 115 (2000), pp. 1–24] about minimal dynamical systems non-embeddable in
[
0
,
1
]
Z
[0,1]^{\mathbb {Z}}
.