Convex set

 ( Convex Analysis ) 

In geometry, a set of points is convex if it contains every line segment between two points in the set.

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 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.

Spaces in which convex sets are defined include the , the over the , and certain non-Euclidean geometries.

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Definitions Let be a vector space or an affine space over the , 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 in and in the interval . This implies that convexity is invariant under affine transformations. Further, it implies that a convex set in a real number or complex number topological vector space is path-connected (and therefore also connected space).

A set is if every point on the line segment connecting and other than the endpoints is inside the topological interior of . A closed convex subset is strictly convex if and only if every one of its boundary points is an extreme point.

A set is absolutely convex if it is convex and balanced set.

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Examples The convex of (the set of real numbers) are the intervals and the points of . Some examples of convex subsets of the Euclidean plane are solid , solid triangles, and intersections of solid triangles. Some examples of convex subsets of a Euclidean space are the Archimedean solids and the . The Kepler-Poinsot polyhedra are examples of non-convex sets.

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Non-convex set 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,

Jeffrey J. McConnell (2025). 9780763722500, Jones & Bartlett Learning. . ISBN 9780763722500

. and some sources more generally use the term concave set to mean a non-convex set, but most authorities prohibit this usage.

Akira Takayama (1994). 9780472081356, University of Michigan Press. . ISBN 9780472081356

(2025). 9781400833085, Princeton University Press. . ISBN 9781400833085

The complement 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..

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Properties Given points in a convex set , and negative number such that , the affine combination $$\backslash sum\_\{k=1\}^r\backslash lambda\_k\; u\_k$$ 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 . The convex hull of a subset of a real vector space is defined as the intersection of all convex sets that contain . More concretely, the convex hull is the set of all convex combinations of points in . In particular, this is a convex set.

A (bounded) convex polytope is the convex hull of a finite subset of some Euclidean space .

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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).

Ivan Singer (1997). 9780471160151, John Wiley & Sons, Inc.. ISBN 9780471160151

1. The empty set and the whole space are convex.
2. The intersection of any collection of convex sets is convex.
3. The union of a collection of convex sets is convex if those sets form a chain (a totally ordered set) under inclusion. For this property, the restriction to chains is important, as the union of two convex sets need not be convex.

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Closed convex sets closed set 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 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.

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Face of a convex set A face of a convex set $C$ is a convex subset $F$ of $C$ such that whenever a point $p$ in $F$ lies strictly between two points $x$ and $y$ in $C$, both $x$ and $y$ must be in $F$. Equivalently, for any $x,y\backslash in\; C$ and any real number trivial faces of $C$. An extreme point of $C$ is a point that is a face of $C$.

Let $C$ be a convex set in $\backslash R^n$ that is compact space (or equivalently, closed and bounded set). Then $C$ is the convex hull of its extreme points. More generally, each compact convex set in a locally convex topological vector space is the closed convex hull of its extreme points (the Krein–Milman theorem).

For example:

• A triangle in the plane (including the region inside) is a compact convex set. Its nontrivial faces are the three vertices and the three edges. (So the only extreme points are the three vertices.)
• The only nontrivial faces of the closed unit disk $\backslash \{\; (x,y)\; \backslash in\; \backslash R^2:\; x^2+y^2\; \backslash leq\; 1\; \backslash \}$ are its extreme points, namely the points on the unit circle $S^1\; =\; \backslash \{\; (x,y)\; \backslash in\; \backslash R^2:\; x^2+y^2=1\; \backslash \}$.

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Convex sets and rectangles Let be a convex body in the plane (a convex set whose interior is non-empty). We can inscribe a rectangle r in such that a homothetic copy R of r is circumscribed about . The positive homothety ratio is at most 2 and: $$\backslash tfrac\{1\}\{2\}\; \backslash cdot\backslash operatorname\{Area\}(R)\; \backslash leq\; \backslash operatorname\{Area\}(C)\; \backslash leq\; 2\backslash cdot\; \backslash operatorname\{Area\}(r)$$

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Blaschke-Santaló diagrams The set $\backslash mathcal\{K\}^2$ 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 $$2r\; \backslash le\; D\; \backslash le\; 2R$$ $$R\; \backslash le\; \backslash frac\{\backslash sqrt\{3\}\}\{3\}\; D$$ $$r\; +\; R\; \backslash le\; D$$ $$D^2\; \backslash sqrt\{4R^2-D^2\}\; \backslash le\; 2R\; (2R\; +\; \backslash sqrt\{4R^2\; -D^2\})$$ 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 $\backslash mathcal\{K\}^2$ can also be parametrized by its width (the smallest distance between any two different parallel support hyperplanes), perimeter and area.

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Other properties

Let X be a topological vector space and $C\; \backslash subseteq\; X$ be convex.

• $\backslash operatorname\{Cl\}\; C$ and $\backslash operatorname\{Int\}\; C$ are both convex (i.e. the closure and interior of convex sets are convex).
• If $a\; \backslash in\; \backslash operatorname\{Int\}\; C$ and $b\; \backslash in\; \backslash operatorname\{Cl\}\; C$ then $[a,\; b[\; \backslash ,\; \backslash subseteq\; \backslash operatorname\{Int\}\; C$ (where ).
• If $\backslash operatorname\{Int\}\; C\; \backslash neq\; \backslash emptyset$ then:
• $\backslash operatorname\{cl\}\; \backslash left(\; \backslash operatorname\{Int\}\; C\; \backslash right)\; =\; \backslash operatorname\{Cl\}\; C$, and
• $\backslash operatorname\{Int\}\; C\; =\; \backslash operatorname\{Int\}\; \backslash left(\; \backslash operatorname\{Cl\}\; C\; \backslash right)\; =\; C^i$, where $C^\{i\}$ is the algebraic interior of C.

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Convex hulls and Minkowski sums

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Convex hulls

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 closure operator:

• extensive: ,
• non-decreasing: implies that , and
• idempotence: .

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 $$\backslash operatorname\{Conv\}(S)\backslash vee\backslash operatorname\{Conv\}(T)\; =\; \backslash operatorname\{Conv\}(S\backslash cup\; T)\; =\; \backslash operatorname\{Conv\}\backslash bigl(\backslash operatorname\{Conv\}(S)\backslash cup\backslash operatorname\{Conv\}(T)\backslash bigr).$$ 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.

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Minkowski addition

In a real vector-space, the Minkowski sum of two (non-empty) sets, and , is defined to be the sumset formed by the addition of vectors element-wise from the summand-sets $$S\_1+S\_2=\backslash \{x\_1+x\_2:\; x\_1\backslash in\; S\_1,\; x\_2\backslash in\; S\_2\backslash \}.$$ More generally, the Minkowski sum of a finite family of (non-empty) sets is the set formed by element-wise addition of vectors $$\backslash sum\_n\; S\_n\; =\; \backslash left\; \backslash \{\; \backslash sum\_n\; x\_n\; :\; x\_n\; \backslash in\; S\_n\; \backslash right\; \backslash \}.$$

For Minkowski addition, the zero set  containing only the null vector  has identity element: For every non-empty subset S of a vector space $$S+\backslash \{0\backslash \}=S;$$ in algebraic terminology, is the identity element of Minkowski addition (on the collection of non-empty sets).The empty set is important in Minkowski addition, because the empty set annihilates every other subset: For every subset of a vector space, its sum with the empty set is empty: $S+\backslash emptyset=\backslash emptyset$.

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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 $$\backslash operatorname\{Conv\}(S\_1+S\_2)=\backslash operatorname\{Conv\}(S\_1)+\backslash operatorname\{Conv\}(S\_2).$$

This result holds more generally for each finite collection of non-empty sets: $$\backslash text\{Conv\}\backslash left\; (\; \backslash sum\_n\; S\_n\; \backslash right\; )\; =\; \backslash sum\_n\; \backslash text\{Conv\}\; \backslash left\; (S\_n\; \backslash right).$$

In mathematical terminology, the operations of Minkowski summation and of forming are commutativity operations.Theorem 3 (pages 562–563): For the commutativity of Minkowski addition and convex hull, see Theorem 1.1.2 (pages 2–3) in Schneider; this reference discusses much of the literature on the of Minkowski in its "Chapter 3 Minkowski addition" (pages 126–196):

Rolf Schneider (1993). 9780521352208, Cambridge University Press. . ISBN 9780521352208

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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.Lemma 5.3:

(2025). 9783540295877, Springer. ISBN 9783540295877

The following famous theorem, proved by Dieudonné in 1966, gives a sufficient condition for the difference of two closed convex subsets to be closed.

C. Zălinescu (2025). 9789812380678, World Scientific Publishing Co., Inc. . ISBN 9789812380678

It uses the concept of a recession cone of a non-empty convex subset S, defined as: $$\backslash operatorname\{rec\}\; S\; =\; \backslash left\backslash \{\; x\; \backslash in\; X\; \backslash ,\; :\; \backslash ,\; x\; +\; S\; \backslash subseteq\; S\; \backslash right\backslash \},$$ where this set is a convex cone containing $0\; \backslash in\; X$ and satisfying $S\; +\; \backslash operatorname\{rec\}\; S\; =\; S$. Note that if S is closed and convex then $\backslash operatorname\{rec\}\; S$ is closed and for all $s\_0\; \backslash in\; S$, $$\backslash operatorname\{rec\}\; S\; =\; \backslash bigcap\_\{t\; >\; 0\}\; t\; (S\; -\; s\_0).$$

Theorem (Dieudonné). Let A and B be non-empty, closed, and convex subsets of a locally convex topological vector space such that $\backslash operatorname\{rec\}\; A\; \backslash cap\; \backslash operatorname\{rec\}\; B$ is a linear subspace. If A or B is locally compact then A −  B is closed.

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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.

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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.

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Orthogonal convexity

An example of generalized convexity is orthogonal convexity.Rawlins G.J.E. and Wood D, "Ortho-convexity and its generalizations", in: Computational Morphology, 137-152. Elsevier, 1988.

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.

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Non-Euclidean geometry

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 joining any two points in the set.

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Order topology

Convexity can be extended for a totally ordered set endowed with the order topology.James Munkres; 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 connected in general: a counter-example is given by the subspace {1,2,3} in , which is both convex and not connected.

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Convexity spaces

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

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

Marcel L. J. van De Vel (1993). 9780444815057, North-Holland Publishing Co.. ISBN 9780444815057

1. The empty set and are in .
2. The intersection of any collection from is in .
3. The union of a Total order (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 .

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Convex spaces Convexity can be generalised as an abstract algebraic structure: a space is convex if it is possible to take convex combinations of points.

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See also

• Absorbing set
• Algorithmic problems on convex sets
• Bounded set (topological vector space)
• Brouwer fixed-point theorem
• Complex convexity
• Convex cone
• Convex series
• Convex metric space
• Carathéodory's theorem (convex hull)
• Choquet theory
• Helly's theorem
• Holomorphically convex hull
• Integrally-convex set
• John ellipsoid
• Pseudoconvexity
• Radon's theorem
• Shapley–Folkman lemma
• Symmetric set

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Bibliography

• R. T. Rockafellar (1997). 9781400873173, Princeton University Press. . ISBN 9781400873173

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External links

• Lectures on Convex Sets, notes by Niels Lauritzen, at Aarhus University, March 2010.

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