  We show that the space of open subsets of any complete and exact symplectic
$4$-manifold has infinite dimension with respect to the symplectic Banach-Mazur
distance; the quasi-flats we construct take values in the set of dynamically
convex domains. In the case of $\mathbb{R}^4$, we therefore obtain the
following contrast: the space of convex domains is quasi-isometric to a plane,
while the space of dynamically convex ones has infinite dimension.
  In the case of $T^* S^2$, a variant of our construction resolves a conjecture
of Stojisavljevi\'{c} and Zhang, asserting that the space of star-shaped
domains in $T^* S^2$ has infinite dimension. Another corollary is that the
space of contact forms giving the standard contact structure on $S^3$ has
infinite dimension with respect to the contact Banach-Mazur distance.
