Illustration of a non-convex set. Since the (red) part of the (black and red) line-segment joining the points x and y lies ''outside'' of the (green) set, the set is non-convex.
, a subset of a Euclidean space
, or more generally an affine space
over the reals
, is convex if, given any two points, it contains the whole line segment
that joins them. Equivalently, a convex set or a convex region is a subset that intersect every line
into a single line segment (possibly empty).
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.
of a convex set is always a convex curve
. The intersection of all the convex sets that contain a given subset of Euclidean space is called the convex hull
of . It is the smallest convex set containing .
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
The notion of a convex set can be generalized as described below.
Let be a vector space
or an affine space
over the real number
s, 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
. This implies that convexity (the property of being convex) is invariant under affine transformation
s. This implies also that a convex set in a real
or complex topological vector space
, thus connected
A set is ' if every point on the line segment connecting and other than the endpoints is inside the interior
A set is ''absolutely convex
'' if it is convex and balanced
The convex subset
s 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 polygon
s, solid triangles, and intersections of solid triangles. Some examples of convex subsets of a Euclidean 3-dimensional space
are the Archimedean solid
s and the Platonic solid
s. The Kepler-Poinsot polyhedra
are examples of non-convex sets.
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
, and some sources more generally use the term ''concave set'' to mean a non-convex set, but most authorities prohibit this usage.
of a convex set, such as the epigraph
of a concave function
, is sometimes called a ''reverse convex set'', especially in the context of mathematical optimization
Given points in a convex set , and
s 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
Intersections and unions
The 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.
'' 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
convex sets are convex sets that contain all their limit points
. They can be characterised as the intersections of ''closed half-space
s'' (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 rectangles
Let ''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
copy ''R'' of ''r'' is circumscribed about ''C''. The positive homothety ratio is at most 2 and:
of all planar convex bodies can be parameterized in terms of the convex body 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.
Let ''X'' be a topological vector space and
are both convex (i.e. the closure and interior of convex sets are convex).
is the [[algebraic interior of ''C''.
Convex hulls and Minkowski sums
Every 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 hull operator
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, and , is defined to be the 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 zero vector
has 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 sums
Minkowski 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
s of Minkowski summation and of forming convex hull
s are commuting
[For the commutativity of Minkowski addition and convexification, see Theorem 1.1.2 (pages 2–3) in Schneider; this reference discusses much of the literature on the convex hulls of Minkowski sumsets in its "Chapter 3 Minkowski addition" (pages 126–196): ]
Minkowski sums of convex sets
The 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
. 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
is a linear subspace. If ''A'' or ''B'' is locally compact
then ''A'' − ''B'' is closed.
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.
Star-convex (star-shaped) sets
Let 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.
An 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.
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 geodesic
s joining any two points in the set.
Convexity can be extended for a totally ordered set
endowed with the order topology
[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.
The notion of convexity may be generalised to other objects, if certain properties of convexity are selected as axiom
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 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 antimatroid
* 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
* Radon's theorem
* Shapley–Folkman lemma
* Symmetric set
Lectures on Convex Sets
notes by Niels Lauritzen, at Aarhus University
, March 2010.