On certain quantifications of Gromov’s nonsqueezing theorem

Let R > 1 and let B be the Euclidean 4-ball of radius R with a closed subset E removed.Suppose that B embeds symplectically into the unit cylinder D 2 × R 2 .By Gromov's non-squeezing theorem, E must be non-empty.We prove that the Minkowski dimension of E is at least 2, and we exhibit an explicit example showing that this result is optimal at least for R ≤ √ 2. In an appendix by Joé Brendel, it is shown that the lower bound is optimal for R < √ 3. We also discuss the minimum volume of E in the case that the symplectic embedding extends, with bounded Lipschitz constant, to the entire ball.