# convex hull of $S$ is open if $S$ is open

Theorem If $S$ is an open set in a topological vector space, then the convex hull $\operatorname{co}(S)$ is open.

As the next example shows, the corresponding result does not hold for a closed set.

Example (Valentine, p. 14) If

 $S=\{(x,1/|x|)\in\mathbbmss{R}^{2}\mid x\in\mathbbmss{R}\setminus\{0\}\},$

then $S$ is closed, but $\operatorname{co}(S)$ is the open half-space $\{(x,y)\mid x\in\mathbbmss{R},y\in(0,\infty)\}$, which is not closed (points on the $x$-axis are accumulation points not in the set, or also can be seen by checking the complement is not open). $\Box$

Reference
F.A. Valentine, Convex sets, McGraw-Hill book company, 1964.

Title convex hull of $S$ is open if $S$ is open ConvexHullOfSIsOpenIfSIsOpen 2013-03-22 13:44:47 2013-03-22 13:44:47 drini (3) drini (3) 9 drini (3) Theorem msc 47L07 msc 46A55