I need to show that this is true for any dimension $d\ge1$.
So far I can show it for triangle and rectangle in $R2$ and for cube in $R3$. But I fail to show it for circle, and sphere. I even think that it is impossible.
If it is possible I think there is Barsuk or Hadwiger's theorem is used. But I can't figure out how.
Can you please help me to prove it for dimension $d\ge1$?
EDIT: Circle and sphere turn out not to be polytope, my mistake.
But is there a mathematical way to show that polytopes of dimension $d$ can be divided into $2$ parts with smaller diameter?