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

TheoremIf $S$ is an open set in a topological vector space, thenthe 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.