Convex hull possesses only integer extreme points

Question

I have the following question. Consider given natural numbers $ 1 \le l_m <\ldots < l_1 < L $. Is it possible to prove that the convex hull of $ \left\lbrace a \in \mathbb{Z}^m_{\ge 0} \, \middle| \, \sum_{i=1}^m l_ia_i \ge L \right\rbrace $ possesses only integer extreme points? If so, what would be a good idea?

Answer 1

By the definition of the convex hull it holds that given a set of points $V = \{v_1,\ldots, v_m\}$ then the extreme points of $conv(V)$ are a subset of $V$. Using that if follows that the extreme points of $conv(\{a \in \mathbb{Z}^m_{\leq0}\ |\ \sum_{i=1}^m l_i a_i \geq L \})$ are all integral vectors.