A closed convex subset is strictly convex if and only if every one of its boundary points is an extreme point. Further, it implies that a convex set in a real or complex topological vector space is path-connected (and therefore also connected).Ī set C is strictly convex if every point on the line segment connecting x and y other than the endpoints is inside the topological interior of C. This implies that convexity is invariant under affine transformations. This means that the affine combination (1 − t) x + ty belongs to C for all x,y in C and t in the interval. A subset C of S is convex if, for all x and y in C, the line segment connecting x and y is included in C. Let S be a vector space or an affine space over the real numbers, or, more generally, over some ordered field (this includes Euclidean spaces, which are affine spaces). The notion of a convex set can be generalized as described below.ĭefinitions A function is convex if and only if its epigraph, the region (in green) above its graph (in blue), is a convex set. The branch of mathematics devoted to the study of properties of convex sets and convex functions is called convex analysis. Convex minimization is a subfield of optimization that studies the problem of minimizing convex functions over convex sets. It is the smallest convex set containing A.Ī 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. The intersection of all the convex sets that contain a given subset A of Euclidean space is called the convex hull of A. The boundary of a convex set is always a convex curve. įor 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. Equivalently, a convex set or a convex region is a subset that intersects every line into a single line segment (possibly empty). In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. The line segment joining points x and y partially extends outside of the set, illustrated in red, and the intersection of the set with the line occurs in two places, illustrated in black. Since this is true for any potential locations of two points within the set, the set is convex. The line segment joining points x and y lies completely within the set, illustrated in green. In geometry, set whose intersection with every line is a single line segment Illustration of a convex set shaped like a deformed circle.
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