Given a convex polytope $P \subset \mathbb R^d$, suppose $p_1, \dots, p_t \in P$. Denote by $\text{Conv}(\{p_1, \dots, p\})$ the convex hull of $\{p_1, \dots, p_t\}$.
Do we have $\text{Conv}(\{p_1, \dots, p_t\}) \subset P$? I think so, because if $p \in \text{Conv}(\{p_1, \dots, p_t\})$, then $p$ is a convex combination of $p_1, \dots, p_t$, so $p = \alpha_1 p_1 + \dots + \alpha_t p_t$, where $\alpha_1, \dots, \alpha_t \geq 0$ and $\sum_{i = 1}^t \alpha_i = 1$.
But then because $p_1, \dots, p_t \in P$, and $P$ is convex, it must contain $\alpha_1 p_1 + \dots + \alpha_t p_t = p$, so $p \in P$, and we are done.
Is this right?