# proof that the convex hull of $S$ is open if $S$ is open

Let $S$ be an open set in some topological vector space $V$. For any sequence of positive real numbers $\Lambda=(\lambda_{1},\ldots,\lambda_{n})$ with $\sum_{i=1}^{n}\lambda_{i}=1$ define

 $S_{\Lambda}=\left\{x\in V\text{ such that }x=\sum_{i=1}^{n}\lambda_{i}s_{i}% \text{ for }s_{i}\in S\right\}.$

Then since addition and scalar multiplication are both open maps, each $S_{\Lambda}$ is open. Finally, the convex hull is clearly just

 $\bigcup_{\Lambda}S_{\Lambda},$

which is therefore open.

Title proof that the convex hull of $S$ is open if $S$ is open ProofThatTheConvexHullOfSIsOpenIfSIsOpen 2013-03-22 14:09:48 2013-03-22 14:09:48 archibal (4430) archibal (4430) 6 archibal (4430) Proof msc 47L07 msc 46A55