Convex set

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 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 $S$ 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 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, $a, b \in S$ and $c \not\in S$ such that $c$ lies between $a$ and $b$. To prove that a set is convex, one must show that no such triple exists.

See Also

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