convex set

Let $S$ a subset of . 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 $$\sum_{k=1}^r\lambda_k u_k$$ 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 or ). 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.

Remark. The notion of convexity can be generalized to an arbitrary partially ordered set: given a poset $P$ (with partial ordering $\le$ ), a subset $C$ of $P$ is said to be convex if for any $a,b\in C$ , if $c\in P$ is between $a$ and $b$ , that is, $a\le c \le b$ , then $c\in C$ .