A note on the wild symplectic ellipsoids

We show that the symplectic $2$-product of $n$ two-dimensional star-shaped domains has an interior symplectomorphic to that of a symplectic ellipsoid. Adapting this construction, given $0<\alpha \leq 1$, we obtain that every open subset of $\mathbb{R}^{2n}$ with a smooth boundary is symplectomorphic to an open set whose boundary contains a set of Hausdorff dimension $2n-1+\alpha$.