I'm doing this exercise and have trouble with the definition of centrally symmetric polytopes.
I understand what it means, but it just doesn't look like a workable definition in solving this problem. I thought of an alternative one, and I'd like to ask if it is equivalent to the one mentioned above.
Definition: A polytope P is said to be centrally symmetric if, for all points $x \in P$, there exists a point $x' \in P$ such that $x+x'=0.$
When you have $x+x'=0$, then with the very same values you would have $x=-x'$, ain't it?
Therefore the whole set $V=\{x, x',...\}$ thus satifies $V=-V$. And as $P=conv(V)$, and therefore $-P=conv(-V)$, you surely have $P=-P$.
--- rk