Convex set

In geometry, a convex set or a convex region is a subset of a Euclidean space, or more generally an affine space over the reals, that intersects every line into a line segment (possibly empty). Equivalently, this is a subset that is closed under convex combinations. 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. In geometry, a convex set or a convex region is a subset of a Euclidean space, or more generally an affine space over the reals, that intersects every line into a line segment (possibly empty). Equivalently, this is a subset that is closed under convex combinations. 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 the convex sets that contain 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.