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]] that 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]]s 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 that are not on the circle. | |
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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]] | ||
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>. | 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>. | ||
| − | A | + | A set in a space that 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 <math>c</math> lies between <math>a</math> and <math>b</math>. To prove that a set is convex, one must show that no such triple exists. |
== See Also == | == See Also == | ||
* [[Convex hull]] | * [[Convex hull]] | ||
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| + | {{stub}} | ||
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| + | [[Category:Geometry]] | ||
Latest revision as of 21:47, 1 March 2008
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 that 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 interiors 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 that 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 set in a space that is not convex is called a concave set. To demonstrate concavity is (in theory) relatively simple: one must find three points, 
and 
such that 
lies between 
and 
. To prove that a set is convex, one must show that no such triple exists.
See Also
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