Consider the open set $M \subset \mathbb{C}^{2}$ given by the union of the unit ball $|z_1|^2 + |z_{2}|^2 < 1$ (the coconut) and the cylinder $|z_1| < \epsilon$, $0 < \epsilon < \! \!< 1$, (the straw, which in this case pierces the coconut through and through, but this is not important).

Fix a number $r$ strictly between $\epsilon$ and $1$.

*Does there exist a strictly positive number $c$ so that the volume of the intersection of the unit ball with any symplectic image of the ball of radius $r$ lying wholly inside $M$ is greater than $c$? If so, is there a reasonable estimate for $c$ as a function of $r$ and $\epsilon$?*

By Gromov's non-squezing theorem we know we cannot symplecticaly move the whole ball of radius $r$ up the straw, but it is not clear to me how much of its *volume* can we sip out of the coconut.

**Motivation.**

This problem is related to the comments I got on this question. Both questions arose from trying to understand a comment that V.I. Arnold made in one of his papers (I forgot where, but for some reason the statement came back to my mind after many years) saying or implying (I read this a long time ago ...) that despite the non-squeezing theorem and the symplectic camel theorem people in statistical mechanics still happily interchange regions of phase space that have the same volume. Brett's answer seems to imply that Arnold's criticism should not be taken too seriously and that the physicists are right: even if you cannot exchange the regions by canonical transformations, you can exchange all of their volume up to an arbitrarily small amount.

On the other hand, it is still possible that there exists a symplectic refinement of Poincaré recurrence.