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We study directional mean dimension of $\mathbb{Z}^k$-actions (where $k$ is apositive integer). On the one hand, we show that there is a$\mathbb{Z}^2$-action whose directional mean dimension (considered as a$[0,+\infty]$-valued function on the torus) is not continuous. On the otherhand, we prove that if a $\mathbb{Z}^k$-action is continuum-wise expansive,then the values of its $(k-1)$-dimensional directional mean dimension arebounded. This is a generalization (with a view towards Meyerovitch andTsukamoto's theorem on mean dimension and expansive multiparameter actions) ofa classical result due to Ma\~n\'e: Any compact metrizable space admitting anexpansive homeomorphism (with respect to a compatible metric) isfinite-dimensional.

Directional mean dimension and continuum-wise expansive $\mathbb{Z}^k$-actions