We fix $n = 2m \geqslant 4$. The closed Euclidean ball of radius $R$ is $B ^ { 2 m } ( R )$ and the symplectic cylinder of radius $R'$ is $Z ^ { 2 m } ( R' ) = \left\{ \left( x _ { 1 } , \ldots , x _ { m } , y _ { 1 } , \ldots , y _ { m } \right) \in \mathbb { R } ^ { 2 m } , \quad x _ { 1 } ^ { 2 } + y _ { 1 } ^ { 2 } \leqslant R'^{ 2 } \right\}$. Prove the linear non-squeezing theorem: For $R > 0$ and $R ^ { \prime } > 0$, there exists $\psi \in \operatorname { Symp } _ { b _ { s } } \left( \mathbb { R } ^ { 2 m } \right)$ such that $\psi \left( B ^ { 2 m } ( R ) \right) \subset Z ^ { 2 m } \left( R ^ { \prime } \right)$ if and only if $R \leqslant R ^ { \prime }$.
We fix $n = 2m \geqslant 4$. The closed Euclidean ball of radius $R$ is $B ^ { 2 m } ( R )$ and the symplectic cylinder of radius $R'$ is $Z ^ { 2 m } ( R' ) = \left\{ \left( x _ { 1 } , \ldots , x _ { m } , y _ { 1 } , \ldots , y _ { m } \right) \in \mathbb { R } ^ { 2 m } , \quad x _ { 1 } ^ { 2 } + y _ { 1 } ^ { 2 } \leqslant R'^{ 2 } \right\}$. Prove the linear non-squeezing theorem: For $R > 0$ and $R ^ { \prime } > 0$, there exists $\psi \in \operatorname { Symp } _ { b _ { s } } \left( \mathbb { R } ^ { 2 m } \right)$ such that $\psi \left( B ^ { 2 m } ( R ) \right) \subset Z ^ { 2 m } \left( R ^ { \prime } \right)$ if and only if $R \leqslant R ^ { \prime }$.