On a question of Dusa McDuff
  Consider the $2n$-dimensional closed ball $B$ of radius 1 in the $2n$-dimensional symplectic cylinder $Z = D \times R^{2n-2}$ over the closed disc $D$ of radius 1. We construct for each $\epsilon >0$ a Hamiltonian deformation $\phi$ of $B$ in $Z$ of energy less than $\epsilon$ such that the area of each intersection of $\phi (B)$ with the disc $D \times \{x\}$, $x \in R^{2n-2}$, is less than $\epsilon$. 
Felix Schlenk