Convex set

Revision as of 09:51, 30 July 2006 by JBL (talk | contribs)

This article is a stub. Help us out by expanding it.

Informally, a convex set $S$ is a set of points such that for any pair of points in the set, all the points between them (that is, on the line segment which joins them) are members of the set as well. Thus, every point in a convex set can "see" every other point in the set. The interior of circles and of all regular polygons are convex, but a circle itself is not because every segment joining two points on the circle contains points which are not on the circle.

More formally, a set $S$ in a space that allows for addition of points and multiplication by real numbers (such as any real vector space) is said to be convex if for any $a,b\in S$ and $0\le t\le 1$, $ta+(1-t)b\in S$.

A region in a space which is not convex is called a concave set. To demonstrate concavity is (in theory) relatively simple: one must find three points, $a, b \in S$ and $c \not\in S$ such that c lies between a and b. To prove that a set is convex, we must show that no such triple exists.

See Also