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convex set (Definition)

Let $S$ a subset of $\mathbbmss{R}^n$. We say that $S$ is convex when, for any pair of points $A,B$ in $S$, the segment $\overline{AB}$ lies entirely inside $S$.

The former statement is equivalent to saying that for any pair of vectors $u,v$ in $S$, the vector $(1-t)u+tv$ is in $S$ for all $t\in[0,1]$.

If $S$ is a convex set, for any $u_1,u_2,\ldots,u_r$ in $S$, and any positive numbers $\lambda_1,\lambda_2,\ldots,\lambda_r$ such that $\lambda_1+\lambda_2+\cdots+\lambda_r=1$ the vector

\begin{displaymath}\sum_{k=1}^r\lambda_k u_k\end{displaymath}

is in $S$.


Examples of convex sets in the plane are circles, triangles, and ellipses. The definition given above can be generalized to any real vector space:

Let $V$ be a vector space (over $\mathbbmss{R}$ or $\mathbbmss{C}$). A subset $S$ of $V$ is convex if for all points $x,y$ in $S$, the line segment $\{\alpha x + (1-\alpha) y \mid \alpha\in(0,1)\} $ is also in $S$.

More generally, the same definition works for any vector space over an ordered field.

A polyconvex set is a finite union of compact, convex sets.



"convex set" is owned by drini. [ full author list (3) | owner history (1) ]
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See Also: convex combination, Carathéodory's theorem, extreme subset of convex set, properties of extreme subsets of a closed convex set

Other names:  convex
Also defines:  polyconvex set, polyconvex

Attachments:
if $A$ is convex and $f$ linear then $f(A)$ and $f^{-1}(A)$ are convex (Theorem) by matte
face of a convex set (Definition) by CWoo
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Cross-references: compact, union, finite, ordered field, line segment, vector space, real, ellipses, triangles, circles, plane, positive, vectors, equivalent, segment, points, subset
There are 94 references to this entry.

This is version 14 of convex set, born on 2001-10-15, modified 2007-06-18.
Object id is 243, canonical name is ConvexSet.
Accessed 20032 times total.

Classification:
AMS MSC: 52A99 (Convex and discrete geometry :: General convexity :: Miscellaneous)
Pending Errata and Addenda
None.
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Discussion
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some technical points by drini on 2003-11-23 02:35:07
when I mention "circle" I mean to include also the interior.
Somebody mention circle refers only to the boundary which is a valid point, since that meaning is sometimes used, but a rather precise way to refer ONLY to the boundary is circumference

The same remarks about including the interior points apply to triangle, ellipses, etc.

It was also suggested to use "ball" instead "circle" but I think the term ball has some topological connotations (indeed, psychological, not mathematical) and whereas on the standard topology balls are circle-shaped, that might not be true on every possible metric that R^n can be given

And just to be even more precise, when I said "interior" in the first two paragraphs I mean interior in the common sense way (i.e., inside points) not interior in a topological way

 f
G -----> H G
p \ /_ ----- ~ f(G)
 \ / f ker f
 G/ker f 
[ reply | up ]
clearing by drini on 2002-02-21 00:24:24
I meant R^n which is the standard way to denote an n-th dimensional euclidean space...
(as akrowne point cartesian product )
and convex sets not necessarily live on the plane (R^2)

:) I see you have lot of issues with standard mathematical notation
 f
G -----> H G
p \ /_ ----- ~ f(G)
 \ / f ker f
 G/ker f 
[ reply | up ]
In R^n? by Logan on 2001-10-17 15:50:57
Should that not be in R^{n\times n}, since it is on the plane?

Logan
[ reply | up ]
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