I generate 1000 polytopes $P_1, \ldots, P_{1000}$ in $\mathbb R^{n}$, each of them has $m$ vertices that are $m$ rows of an $m\times n$ matrix $A_i={\sf rand(m,n)}$. Then I take their sum $P=P_1+\ldots +P_{1000}$. From what I have observed by plotting $P$, it seems that $P$ is nearly round.
Can you prove this fact or can you give an example in which the sum of 1000 polytopes (which has nonempty interior) is really flat.