I have a simple question that appeared in research: For a rectangle $S :=[a_1,b_1] \times[a_2,b_2] \times \dots \times [a_n,b_n] \subset \mathbb{R}^n$. Let $p_0 = (a_1,a_2,\dots,a_n)$, and define $p_i (1 \leq i\leq n)$ be the vector obtained by changing the $i$th entry of $p_0$ to $b_i$.

I think $\operatorname{cone}(S) = \operatorname{cone}(\{ p_i\mid 0 \leq i\leq n \})$, but I'm not sure if this is true.

Intuitively, I suspect this is true. I tried to use Farkas Lemma on this but did not reach anything promising.

I asked my professor about this and he went on talking about $\mu$ analysis etc, and saying something about capturing the set $\{ A + B\triangle C\mid \|\triangle\|_\text{op} \leq 1 \}$. However, I felt my question is very straightforward, and does not require much advanced tools

My motivation of this problem is about simultaneous stability, where all the corners are $n \times n$ matrix, so checking $2^n$ corners is computationally infeasible.

This is my first post on Mathoverflow, previously I have mostly posted on Mathematics stack exchange, but according to this post, since this is research related and might get more attention here, I decided to post here.