Difference between revisions of "Convex set"
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| + | Informally, a '''convex''' [[set]] <math>S</math> is a set of [[point]]s 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 the set as well. Thus, every point in a convex set can "see" every other point in the set. The [[interior]] of [[circle]]s and of all [[regular polygon]]s 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. | ||
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| + | More formally, a set <math>S</math> in a space that allows for [[addition]] of points and [[multiplication]] by [[real number]]s (such as any real [[vector space]]) is said to be '''convex''' if for any <math>a,b\in S</math> and <math>0\le t\le 1</math>, <math>ta+(1-t)b\in S</math>. | ||
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| + | 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, <math>a, b \in S</math> and <math>c \not\in S</math> 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 == | == See Also == | ||
| − | * [[ | + | * [[Convex hull]] |
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Revision as of 10:07, 13 July 2006
This article is a stub. Help us out by expanding it.
Informally, a convex set 
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 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 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 
and 
, 
.
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, 
and 
such that c lies between a and b. To prove that a set is convex, we must show that no such triple exists.
