Let $S_1, \dots, S_k \subset \mathbb{R}^n$ be a set of (non-regular) simplices. Let $m_i$ indicate the number of vertices of simplex $S_i$ (we do not assume it is equal to $n-1$).

Is there a simple upper bound on the maximum number of vertices of the intersection $\bigcap_i S_i$, stated in terms of the set $\{ m_i\}_{i=1..k}$?

What about the maximum number of vertices of the convex hull of the union, $\mathrm{Conv}\big(\bigcup_i S_i\big)$?

(Cross-posting from math.SE)