Let $P$ be a convex polytope in $\Bbb R^n$ with the origin in its interior such that for all vertices $x$ of $P$ we have $||x||_2 \geq 3$. Then at least one facet of $P$ does not intersect the Euclidean unit ball $B_2^n=\{x\in\Bbb R^n: ||x||_2\leq 1\}$.
Do you think that this is true? If so, can anyone provide a strategy to prove this? If, in order to make the conclusion hold, one needs to assume that all vertices satisfy $||x||_2 \geq c$, where $c>3$ is a universal constant (independent of the dimension $n$ and the polytope $P$), that is fine.
Thank you in advance.
I think this is false: consider, for large $n$, the hypercube with vertices $(3/\sqrt{n})(\pm1,\pm1,\ldots,\pm1)$.