Convex set

In a Euclidean space (or, more generally in an affine space), a convex set is a region such that, for every pair of points within the region, every point on the straight line segment that joins the pair of points is also within the region. For example, a solid cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex.

The boundary of a convex set is always a convex curve. The intersection of all convex sets containing a given subset A of Euclidean space is called the convex hull of A. It is the smallest convex set containing A.

A convex function is a real-valued function defined on an interval with the property that its epigraph (the set of points on or above the graph of the function) is a convex set. Convex minimization is a subfield of optimization that studies the problem of minimizing convex functions over convex sets. The branch of mathematics devoted to the study of properties of convex sets and convex functions is called convex analysis.

The notion of a convex set can be generalized to other spaces as described below.

Let S be a vector space over the real numbers, or, more generally, some ordered field. This includes Euclidean spaces. A set C in S is said to be convex if, for all x and y in C and all t in the interval [0, 1], the point (1 − t)x + ty also belongs to C. In other words, every point on the line segment connecting x and y is in C. This implies that a convex set in a real or complex topological vector space is path-connected, thus connected. Furthermore, C is strictly convex if every point on the line segment connecting x and y other than the endpoints is inside the interior of C.

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