In convex geometry, a convex set is a subset of an affine space that is closed under convex combinations. For example, a cube is a convex set, but anything that is hollow or has an indent, for example. The boundary of a set is always a convex curve. The intersection of all convex sets containing a given subset A of Euclidean space is called the 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 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 set can be generalized as described below. Let S be a space over the real numbers, or, more generally. A set C in S is said to be if, for all x and y in C and all t in the interval. In other words, every point on the segment connecting x and y is in C. This implies that a set in a real or complex topological vector space is path-connected. Furthermore, C is strictly convex if every point on the segment connecting x and y other than the endpoints is inside the interior of C. A set C is called convex if it is convex. The convex subsets of R are simply the intervals of R, some examples of convex subsets of the Euclidean plane are solid regular polygons, solid triangles, and intersections of solid triangles. Some examples of convex subsets of a Euclidean 3-dimensional space are the Archimedean solids, the Kepler-Poinsot polyhedra are examples of non-convex sets. A set that is not convex is called a non-convex set, the complement of a convex set, such as the epigraph of a concave function, is sometimes called a reverse convex set, especially in the context of mathematical optimization. If S is a set in n-dimensional space, then for any collection of r, r >1. Ur in S, and for any nonnegative numbers λ1, + λr =1, then one has, ∑ k =1 r λ k u k ∈ S
Illustration of a convex set which looks somewhat like a deformed circle. The (black) line segment joining points x and y lies completely within the (green) set. Since this is true for any points x and y within the set that we might choose, the set is convex.
Illustration of a non-convex set. Since the red part of the (black and red) line-segment joining the points x and y lies outside of the (green) set, the set is non-convex.
A function is convex if and only if its epigraph, the region (in green) above its graph (in blue), is a convex set.