DefinitionsLet 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. A subset of is convex if, for all and in , the line segment connecting and is included in . This means that the affine combination belongs to , for all and in , and in the interval (mathematics), interval . This implies that convexity (the property of being convex) is invariant under affine transformations. This implies also that a convex set in a real number, real or complex number, complex topological vector space is path-connected, thus connected space, connected. A set is ' if every point on the line segment connecting and other than the endpoints is inside the Interior (topology), interior of . A set is ''absolutely convex'' if it is convex and balanced set, balanced. The convex subsets of (the set of real numbers) are the intervals and the points of . 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 space, Euclidean 3-dimensional space are the Archimedean solids and the Platonic solids. The Kepler-Poinsot polyhedra are examples of non-convex sets.
Non-convex setA set that is not convex is called a ''non-convex set''. A polygon that is not a convex polygon is sometimes called a concave polygon, and some sources more generally use the term ''concave set'' to mean a non-convex set, but most authorities prohibit this usage. The Complement (set theory), complement of a convex set, such as the epigraph (mathematics), epigraph of a concave function, is sometimes called a ''reverse convex set'', especially in the context of mathematical optimization.
PropertiesGiven points in a convex set , and negative number, nonnegative numbers such that , the affine combination : belongs to . As the definition of a convex set is the case , this property characterizes convex sets. Such an affine combination is called a convex combination of .
Intersections and unionsThe collection of convex subsets of a vector space, an affine space, or a Euclidean space has the following properties:Soltan, Valeriu, ''Introduction to the Axiomatic Theory of Convexity'', Ştiinţa, Chişinău, 1984 (in Russian). #The empty set and the whole space are convex. #The intersection of any collection of convex sets is convex. #The ''union (sets), union'' of a sequence of convex sets is convex, if they form a Total order#Chains, non-decreasing chain for inclusion. For this property, the restriction to chains is important, as the union of two convex sets ''need not'' be convex.
Closed convex setsclosed set, Closed convex sets are convex sets that contain all their limit points. They can be characterised as the intersections of ''closed Half-space (geometry), half-spaces'' (sets of point in space that lie on and to one side of a hyperplane). From what has just been said, it is clear that such intersections are convex, and they will also be closed sets. To prove the converse, i.e., every closed convex set may be represented as such intersection, one needs the supporting hyperplane theorem in the form that for a given closed convex set and point outside it, there is a closed half-space that contains and not . The supporting hyperplane theorem is a special case of the Hahn–Banach theorem of functional analysis.
Convex sets and rectanglesLet ''C'' be a convex body in the plane (a convex set whose interior is non-empty). We can inscribe a rectangle ''r'' in ''C'' such that a Homothetic transformation, homothetic copy ''R'' of ''r'' is circumscribed about ''C''. The positive homothety ratio is at most 2 and: :
Blaschke-Santaló diagramsThe set of all planar convex bodies can be parameterized in terms of the convex body Diameter#Generalizations, diameter ''D'', its inradius ''r'' (the biggest circle contained in the convex body) and its circumradius ''R'' (the smallest circle containing the convex body). In fact, this set can be described by the set of inequalities given by
and can be visualized as the image of the function ''g'' that maps a convex body to the point given by (''r''/''R'', ''D''/2''R''). The image of this function is known a (''r'', ''D'', ''R'') Blachke-Santaló diagram. Alternatively, the set can also be parametrized by its width (the smallest distance between any two different parallel support hyperplanes), perimeter and area.
Other propertiesLet ''X'' be a topological vector space and be convex. * and are both convex (i.e. the closure and interior of convex sets are convex). * If and then (where ). * If then: ** , and ** , where is the algebraic interior of ''C''.
Convex hulls and Minkowski sums
Convex hullsEvery subset of the vector space is contained within a smallest convex set (called the convex hull of ), namely the intersection of all convex sets containing . The convex-hull operator Conv() has the characteristic properties of a closure operator, hull operator: : The convex-hull operation is needed for the set of convex sets to form a lattice (order), lattice, in which the join and meet, "''join''" operation is the convex hull of the union of two convex sets : . The intersection of any collection of convex sets is itself convex, so the convex subsets of a (real or complex) vector space form a complete lattice (order), lattice.
Minkowski additionIn a real vector-space, the ''Minkowski addition, Minkowski sum'' of two (non-empty) sets, and , is defined to be the sumset, set formed by the addition of vectors element-wise from the summand-sets :. More generally, the ''Minkowski sum'' of a finite family of (non-empty) sets is the set formed by element-wise addition of vectors : For Minkowski addition, the ''zero set'' containing only the null vector, zero vector has identity element, special importance: For every non-empty subset S of a vector space :; in algebraic terminology, is the identity element of Minkowski addition (on the collection of non-empty sets).
Convex hulls of Minkowski sumsMinkowski addition behaves well with respect to the operation of taking convex hulls, as shown by the following proposition: Let be subsets of a real vector-space, the convex hull of their Minkowski sum is the Minkowski sum of their convex hulls :. This result holds more generally for each finite collection of non-empty sets: : In mathematical terminology, the operation (mathematics), operations of Minkowski summation and of forming convex hulls are commutativity, commuting operations.For the commutativity of Minkowski addition and convex hull, convexification, see Theorem 1.1.2 (pages 2–3) in Schneider; this reference discusses much of the literature on the convex hulls of Minkowski addition, Minkowski sumsets in its "Chapter 3 Minkowski addition" (pages 126–196):
Minkowski sums of convex setsThe Minkowski sum of two compact convex sets is compact. The sum of a compact convex set and a closed convex set is closed. The following famous theorem, proved by Dieudonné in 1966, gives a sufficient condition for the difference of two closed convex subsets to be closed. It uses the concept of a recession cone of a non-empty convex subset ''S'', defined as: : . where this set is a convex cone containing and satisfying . Note that if ''S'' is closed and convex then is closed and for all , . Theorem (Dieudonné). Let ''A'' and ''B'' be non-empty, closed, and convex subsets of a locally convex topological vector space such that is a linear subspace. If ''A'' or ''B'' is locally compact then ''A'' − ''B'' is closed.
Generalizations and extensions for convexityThe notion of convexity in the Euclidean space may be generalized by modifying the definition in some or other aspects. The common name "generalized convexity" is used, because the resulting objects retain certain properties of convex sets.
Star-convex (star-shaped) setsLet be a set in a real or complex vector space. is star convex (star-shaped) if there exists an in such that the line segment from to any point in is contained in . Hence a non-empty convex set is always star-convex but a star-convex set is not always convex.
Orthogonal convexityAn example of generalized convexity is orthogonal convexity. A set in the Euclidean space is called orthogonally convex or ortho-convex, if any segment parallel to any of the coordinate axes connecting two points of lies totally within . It is easy to prove that an intersection of any collection of orthoconvex sets is orthoconvex. Some other properties of convex sets are valid as well.
Non-Euclidean geometryThe definition of a convex set and a convex hull extends naturally to geometries which are not Euclidean by defining a geodesic convexity, geodesically convex set to be one that contains the geodesics joining any two points in the set.
Order topologyConvexity can be extended for a totally ordered set endowed with the order topology.James Munkres, Munkres, James; ''Topology'', Prentice Hall; 2nd edition (December 28, 1999). . Let . The subspace is a convex set if for each pair of points in such that , the interval is contained in . That is, is convex if and only if for all in , implies . A convex set is not connected in general: a counter-example is given by the subspace in , which is both convex and not connected.
Convexity spacesThe notion of convexity may be generalised to other objects, if certain properties of convexity are selected as axioms. Given a set , a convexity over is a collection of subsets of satisfying the following axioms: #The empty set and are in #The intersection of any collection from is in . #The union of a Total order, chain (with respect to the inclusion relation) of elements of is in . The elements of are called convex sets and the pair is called a convexity space. For the ordinary convexity, the first two axioms hold, and the third one is trivial. For an alternative definition of abstract convexity, more suited to discrete geometry, see the ''convex geometries'' associated with antimatroids.
See also* Absorbing set * Bounded set (topological vector space) * Brouwer fixed-point theorem * Complex convexity * Convex hull * Convex series * Convex metric space * Carathéodory's theorem (convex hull) * Choquet theory * Helly's theorem * Holomorphically convex hull * Integrally-convex set * Pseudoconvexity * Radon's theorem * Shapley–Folkman lemma * Symmetric set