Convex hull: $n+1$ points suffice in dimension $n$?
Andrew Henderson 
Let $S \subseteq \mathbb{R}^n$ for some $n \in \mathbb{N}$. The convex hull of $S$ is defined to be
$$ \left\lbrace \sum_{i=1}^k \lambda_i x_i \mid k \in \mathbb{N}, x_i \in S, \lambda_i \in \mathbb{R}_{\geq 0}, \sum_{i=1}^k \lambda_i = 1 \right\rbrace. $$
It feels intuitively clear to me that one should be able to replace this by
$$ \left\lbrace \sum_{i=1}^{n+1} \lambda_i x_i \mid x_i \in S, \lambda_i \in \mathbb{R}_{\geq 0}, \sum_{i=1}^{n+1} \lambda_i = 1 \right\rbrace, $$ that is, that every point in the convex hull can actually be written as the convex combination of some $n+1$ points from the original set (depending on the chosen point of course). Is there a short argument for this?
$\endgroup$ 21 Answer
$\begingroup$This is Carathéodory's theorem. Wikipedia has a short argument.
$\endgroup$ 1
