convex hull of S is open if S is open

Theorem If S is an open set in a topological vector spaceMathworldPlanetmath, then the convex hull co(S) is open.

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

Example (Valentine, p. 14) If


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

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