We fix $n = 2m \geqslant 4$ and $B ^ { 2 m } ( r ) = \left\{ \left( x _ { 1 } , \ldots , x _ { m } , y _ { 1 } , \ldots , y _ { m } \right) \in \mathbb { R } ^ { 2 m } , \quad x _ { 1 } ^ { 2 } + \cdots + x _ { m } ^ { 2 } + y _ { 1 } ^ { 2 } + \cdots + y _ { m } ^ { 2 } \leqslant r ^ { 2 } \right\}$. Under what necessary and sufficient condition on $r > 0$ does there exist $u$ belonging to $\mathrm { SL } \left( \mathbb { R } ^ { 2 m } \right)$ such that $u \left( B ^ { 2 m } ( 1 ) \right) \subset B ^ { 2 m } ( r )$?
We fix $n = 2m \geqslant 4$ and $B ^ { 2 m } ( r ) = \left\{ \left( x _ { 1 } , \ldots , x _ { m } , y _ { 1 } , \ldots , y _ { m } \right) \in \mathbb { R } ^ { 2 m } , \quad x _ { 1 } ^ { 2 } + \cdots + x _ { m } ^ { 2 } + y _ { 1 } ^ { 2 } + \cdots + y _ { m } ^ { 2 } \leqslant r ^ { 2 } \right\}$. Under what necessary and sufficient condition on $r > 0$ does there exist $u$ belonging to $\mathrm { SL } \left( \mathbb { R } ^ { 2 m } \right)$ such that $u \left( B ^ { 2 m } ( 1 ) \right) \subset B ^ { 2 m } ( r )$?