Gromov non-squeezing theorem

If there is a symplectic embeddingthen $R\le r$. Symplectic volume alone does not imply this radius obstruction.

The symplectic manifold $T^2\times {\mathbb{R}}^2$ contains symplectic balls of arbitrarily large radius. A steep linear Lagrangian graph in ${\mathbb{R}}^4=T^{\ast }{\mathbb{R}}^2$ has a thin neighborhood that injects under the quotient ${\mathbb{R}}^2\to T^2$, while a symplectic base-fiber dilation fits a large round ball into that long thin neighborhood.