In geometry, set whose intersection with every line is a single line segment
Ingeometry, 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. Equivalently, a convex set or a convex region is a subset that intersects every line into a single line segment (possibly empty).[1][2]
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 in the plane 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 hullofA. It is the smallest convex set containing A.
Aconvex 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 as described below.
A set Cisstrictly convex if every point on the line segment connecting x and y other than the endpoints is inside the topological interiorofC. A closed convex subset is strictly convex if and only if every one of its boundary points is an extreme point.[3]
A 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,[4] and some sources more generally use the term concave set to mean a non-convex set,[5] but most authorities prohibit this usage.[6][7]
Given r points u1, ..., ur in a convex set S, and rnonnegative numbersλ1, ..., λr such that λ1 + ... + λr = 1, the affine combination
belongs to S. As the definition of a convex set is the case r = 2, this property characterizes convex sets.
The intersection of any collection of convex sets is convex.
The union of a sequence of convex sets is convex, if they form a 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 sets are convex sets that contain all their limit points. They can be characterised as the intersections of closed half-spaces (sets of points 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 C and point P outside it, there is a closed half-space H that contains C and not P. The supporting hyperplane theorem is a special case of the Hahn–Banach theoremoffunctional analysis.
Let C be a convex body in the plane (a convex set whose interior is non-empty). We can inscribe a rectangle rinC such that a homothetic copy Rofr is circumscribed about C. The positive homothety ratio is at most 2 and:[11]
The set of all planar convex bodies can be parameterized in terms of the convex body diameterD, 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[12][13]
and can be visualized as the image of the function g that maps a convex body to the R2 point given by (r/R, D/2R). The image of this function is known a (r, D, R) Blachke-Santaló diagram.[13]
Alternatively, the set can also be parametrized by its width (the smallest distance between any two different parallel support hyperplanes), perimeter and area.[12][13]
Every subset A of the vector space is contained within a smallest convex set (called the convex hullofA), namely the intersection of all convex sets containing A. The convex-hull operator Conv() has the characteristic properties of a hull operator:
extensive: S ⊆ Conv(S),
non-decreasing: S ⊆ T implies that Conv(S) ⊆ Conv(T), and
The convex-hull operation is needed for the set of convex sets to form a lattice, in which the "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.
In a real vector-space, the Minkowski sum of two (non-empty) sets, S1 and S2, is defined to be the setS1 + S2 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 Sn is the set formed by element-wise addition of vectors
For Minkowski addition, the zero set{0} containing only the zero vector0 has special importance: For every non-empty subset S of a vector space
in algebraic terminology, {0} is the identity element of Minkowski addition (on the collection of non-empty sets).[14]
The Minkowski sum of two compact convex sets is compact. The sum of a compact convex set and a closed convex set is closed.[17]
The following famous theorem, proved by Dieudonné in 1966, gives a sufficient condition for the difference of two closed convex subsets to be closed.[18] 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 ,
Generalizations and extensions for convexity[edit]
The 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.
Let C be a set in a real or complex vector space. Cisstar convex (star-shaped) if there exists an x0inC such that the line segment from x0 to any point yinC is contained in C. Hence a non-empty convex set is always star-convex but a star-convex set is not always convex.
An example of generalized convexity is orthogonal convexity.[19]
A set S in the Euclidean space is called orthogonally convexorortho-convex, if any segment parallel to any of the coordinate axes connecting two points of S lies totally within S. 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.
The definition of a convex set and a convex hull extends naturally to geometries which are not Euclidean by defining a geodesically convex set to be one that contains the geodesics joining any two points in the set.
Let Y ⊆ X. The subspace Y is a convex set if for each pair of points a, binY such that a ≤ b, the interval [a, b] = {x ∈ X | a ≤ x ≤ b} is contained in Y. That is, Y is convex if and only if for all a, binY, a ≤ b implies [a, b] ⊆ Y.
A convex set is not connected in general: a counter-example is given by the subspace {1,2,3} in Z, which is both convex and not connected.
The elements of 𝒞 are called convex sets and the pair (X, 𝒞) 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.
^Takayama, Akira (1994). Analytical Methods in Economics. University of Michigan Press. p. 54. ISBN9780472081356. An often seen confusion is a "concave set". Concave and convex functions designate certain classes of functions, not of sets, whereas a convex set designates a certain class of sets, and not a class of functions. A "concave set" confuses sets with functions.
^ abSoltan, Valeriu, Introduction to the Axiomatic Theory of Convexity, Ştiinţa, Chişinău, 1984 (in Russian).
^ abSinger, Ivan (1997). Abstract convex analysis. Canadian Mathematical Society series of monographs and advanced texts. New York: John Wiley & Sons, Inc. pp. xxii+491. ISBN0-471-16015-6. MR1461544.
^ abSantaló, L. (1961). "Sobre los sistemas completos de desigualdades entre tres elementos de una figura convexa planas". Mathematicae Notae. 17: 82–104.
^The empty set is important in Minkowski addition, because the empty set annihilates every other subset: For every subset S of a vector space, its sum with the empty set is empty: .
^Theorem 3 (pages 562–563): Krein, M.; Šmulian, V. (1940). "On regularly convex sets in the space conjugate to a Banach space". Annals of Mathematics. Second Series. 41 (3): 556–583. doi:10.2307/1968735. JSTOR1968735.
^van De Vel, Marcel L. J. (1993). Theory of convex structures. North-Holland Mathematical Library. Amsterdam: North-Holland Publishing Co. pp. xvi+540. ISBN0-444-81505-8. MR1234493.