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 $\mathrm{co}(S)$ is open.

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

Example (Valentine, p. 14) If

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

then $S$ is closed, but $\mathrm{co}(S)$ is the open half-space $\{(x,y)\mid x\in ℝ,y\in (0,\mathrm{\infty })\}$, 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). $\mathrm{\square }$

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

Title	convex hull of $S$ is open if $S$ is open
Canonical name	ConvexHullOfSIsOpenIfSIsOpen
Date of creation	2013-03-22 13:44:47
Last modified on	2013-03-22 13:44:47
Owner	drini (3)
Last modified by	drini (3)
Numerical id	9
Author	drini (3)
Entry type	Theorem
Classification	msc 47L07
Classification	msc 46A55