Convex set

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Convex set

In Euclidean space

Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object.

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In Euclidean space

Euclidean space

Crescent

In art and symbolism, a crescent is generally the shape produced when a circle disk has a segment of another circle removed from its edge, so that what remains is a shape enclosed by two circular arcs of different diameters which intersect at two points ....
shape, is not convex.

In Euclidean geometry

Let C be a set in a real

Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
or complex

Complex number

In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:...
vector space

Vector space

File:Vector addition ans scaling.pngA vector space is a mathematical structure formed by a collection of vectors: objects that may be Vector addition together and Scalar multiplication by numbers, called scalar s in this context....
. C is said to be convex if, for all x and y in C and all t in the interval

Interval (mathematics)

In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set....
[0,1], the point

x + t y

is in C. In other words, every point on the line segment

Line segment

In geometry, a line segment is a part of a line that is bounded by two end Point , and contains every point on the line between its end points....
connecting x and y is in C. This implies that a convex set in a real

Real number

Complex number

In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:...
topological vector space

Topological vector space

In mathematics, a topological vector space is one of the basic structures investigated in functional analysis. As the name suggests the space blends a topology with the algebraic concept of a vector space....
is path-connected, thus connected

Connected space

In topology and related branches of mathematics, a connected space is a topological space which cannot be represented as the disjoint union of two or more nonempty open subsets....
.

A set C is called absolutely convex if it is convex and balanced

Balanced set

In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space is a Set S so that for all scalars α with |α| ≤ 1...
.

The convex subset

Subset

In mathematics, especially in set theory, a Set A is a subset of a set B if A is "contained" inside B. Notice that A and B may coincide....
s of R (the set of real numbers) are simply the intervals of R. Some examples of convex subsets of Euclidean 2-space

Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
are regular polygon

Regular polygon

A regular polygon is a polygon which is Equiangular polygon and equilateral . Regular polygons may be convex or Star polygon....
s and bodies of constant width

Curve of constant width

In geometry, a curve of constant width is a Convex set planar shape whose width, measured by the distance between two opposite parallellines touching its boundary, is the same regardless of the direction of those two parallel lines....
. Some examples of convex subsets of Euclidean 3-space

Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
are the Archimedean solid

Archimedean solid

In geometry an Archimedean solid is a highly symmetric, semi-regular convex set polyhedron composed of two or more types of regular polygons meeting in identical vertex ....
s and the Platonic solid

Platonic solid

In geometry, a Platonic solid is a convex set polyhedron that is regular polyhedron, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruence regular polygons, with the same number of faces meeting at each vertex....
s. The Kepler-Poinsot polyhedra are examples of non-convex sets.

Properties

If is a convex set, for any in , and any nonnegative numbers such that , then the vector is in . A vector of this type is known as a convex combination

Convex combination

A convex combination is a linear combination of point where all coefficients are non-negative and sum up to 1. All possible convex combinations will be within the convex hull of the given points....
of .

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

Lattice (order)

In mathematics, a lattice is a partially ordered set in which subsets of any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain Axiom identity ....
. This also means that any subset A of the vector space is contained within a smallest convex set (called the convex hull

Convex hull

In mathematics, the convex hull or convex envelope for a Set of points X in a real vector space V is the minimal convex set containing X....
of A), namely the intersection of all convex sets containing A.

Closed

Closed set

In topology and related branches of mathematics, a closed set is a Set whose complement is open set....
convex sets can be characterised as the intersections of closed half-space
Half-space

In geometry, a half-space is either of the two parts into which a plane divides the three-dimensional space. More generally, a half-space is either of the two parts into which a hyperplane divides an affine space....
s (sets of point in space that lie on and to one side of a hyperplane

Hyperplane

A hyperplane is a concept in geometry. It is a higher-dimensional generalization of the concepts of a line in the plane and a plane in 3-dimensional space....
). 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 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 theorem of functional analysis

Functional analysis

Functional analysis is the branch of mathematics, and specifically of mathematical analysis, concerned with the study of vector spaces and operators acting upon them....
.

Star-convex sets

Let C be a set in a real or complex vector space. C is star convex if there exists an in C such that the line segment from to any point y in C is contained in C. Hence a non-empty convex set is always star-convex but a star-convex set is not always convex.

Generalizations and extensions for convexity

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.

Orthogonal convexity

An example of generalized convexity is orthogonal convexity.

A set S in the Euclidean space is called orthogonally convex or orthoconvex, 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.

See "Orthogonal convex hull

Orthogonal convex hull

In Euclidean geometry, a set is defined to be orthogonally convex if, for every line L that is parallel to one of the axes of the Cartesian coordinate system, the intersection of K with L is empty, a point, or a single interval....
" for more.

Non-Euclidean geometry

The definition of a convex set and a convex hull extends naturally to non-Euclidean geometry

Non-Euclidean geometry

In mathematics, non-Euclidean geometry describes hyperbolic geometry and elliptic geometry, which are contrasted with Euclidean geometry. The essential difference between Euclidean and non-Euclidean geometry is the nature of Parallel lines....
by defining a geodesically convex set

Geodesic convexity

In mathematics — specifically, in Riemannian geometry — geodesic convexity is a natural generalization of convex set and convex function to Riemannian manifolds....
to be one that contains the geodesic

Geodesic

In mathematics, a geodesic [jee-uh-des-ik, -dee-sik] is a generalization of the notion of a "Line " to "manifolds".In presence of a Metric , geodesics are defined to be the shortest path between points on the space....
s joining any two points in the set.

Order topology

Convexity can be extended for a space endowed with the order topology

Order topology

In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets....
, using the total order

Total order

In mathematics and set theory, a total order, linear order, simple order, or ordering is a binary relation on some Set X....
of the space.

Let . The subspace is a convex set if for each pair of points such that , the interval is contained in . That is, is convex if and only if .

Abstract (axiomatic) convexity

The notion of convexity may be generalised to other objects, if certain properties of convexity are selected as axiom

Axiom

In traditional logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evidence, or subject to necessary decision....
s.

Given a set X, a convexity over X is a collection of subsets of X satisfying the following axioms:

The empty set and X are in
The intersection of any collection from is in .
The union of a chain
Total order

In mathematics and set theory, a total order, linear order, simple order, or ordering is a binary relation on some Set X....
(with respect to the inclusion relation) of elements of is in .

The elements of are called convex sets and the pair (X, ) 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

Discrete geometry

Discrete geometry or combinatorial geometry may be loosely defined as study of geometrical objects and properties that are discrete space or combinatorial, either by their nature or by their representation; the study that does not essentially rely on the notion of continuum....
, see the convex geometries associated with antimatroid

Antimatroid

In mathematics, an antimatroid is a formal system that describes processes in which a set is built up by including elements one at a time, and in which an element, once available for inclusion, remains available until it is included....
s.

Convex set

In Euclidean geometry

Properties

Star-convex sets

Generalizations and extensions for convexity

Orthogonal convexity

Non-Euclidean geometry

Order topology

Abstract (axiomatic) convexity

See also