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[parent] convex hull of $S$ is open if $S$ is open (Theorem)

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

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

Example (Valentine, p. 14) If

$\displaystyle S=\{ (x,1/\vert x\vert)\in \mathbbmss{R}^2 \mid x\in \mathbbmss{R}\setminus \{0\} \},$
then $S$ is closed, but $\cull (S)$ 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.



"convex hull of $S$ is open if $S$ is open" is owned by drini. [ full author list (2) | owner history (2) ]
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proof that the convex hull of $S$ is open if $S$ is open (Proof) by archibal
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Cross-references: convex sets, reference, complement, accumulation points, points, closed, closed set, open, convex hull, topological vector space, open set, theorem

This is version 6 of convex hull of $S$ is open if $S$ is open, born on 2003-07-13, modified 2004-02-18.
Object id is 4443, canonical name is ConvexCombinationOfSIfOpenIfSIsOpen2.
Accessed 2400 times total.

Classification:
AMS MSC: 47L07 (Operator theory :: Linear spaces and algebras of operators :: Convex sets and cones of operators)
 46A55 (Functional analysis :: Topological linear spaces and related structures :: Convex sets in topological linear spaces; Choquet theory)
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