 Convex set

In Euclidean 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. For example, a solid cube is convex, but anything that is hollow or has a dent in it, for example, a crescent shape, is not convex.
The notion can be generalized to other spaces as described below.
Contents
In vector spaces
Let S be a vector space over the real numbers, or, more generally, some ordered field. This includes Euclidean spaces. A set C in S is said to be convex if, for all x and y in C and all t in the interval [0,1], the point
 (1 − t ) x + t y
is in C. In other words, every point on the line segment connecting x and y is in C. This implies that a convex set in a real or complex topological vector space is pathconnected, thus connected.
A set C is called absolutely convex if it is convex and balanced.
The convex subsets of R (the set of real numbers) are simply the intervals of R. Some examples of convex subsets of Euclidean 2space are regular polygons and bodies of constant width. Some examples of convex subsets of Euclidean 3space are the Archimedean solids and the Platonic solids. The KeplerPoinsot polyhedra are examples of nonconvex sets.
Properties
If S is a convex set, for any in S, and any nonnegative numbers such that , then the vector is in S. A vector of this type is known as a convex combination of .
Intersections and unions
The collection of convex subsets of a vector space has the following properties:^{[1]}^{[2]}
 The empty set and the whole vectorspace are convex.
 The intersection of any collection of convex sets is convex.
 The union of a nondecreasing sequence of convex subsets is a convex set.
For the preceding property of unions of nondecreasing sequences of convex sets, the restriction to nested sets was important: The union of two convex sets need not be convex.
Convex hulls
Every subset A of the vector space is contained within a smallest convex set (called the convex hull of A), namely the intersection of all convex sets containing A. The convexhull operator Conv() has the characteristic properties of a hull operator:

extensive S ⊆ Conv(S), nondecreasing S ⊆ T implies that Conv(S) ⊆ Conv(T), and idempotent Conv(Conv(S)) = Conv(S).
The convexhull 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
 Conv(S)∨Conv(T) = Conv( S ∪ T ) = Conv( Conv(S) ∪ Conv(T) ).
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.
Minkowski addition
Main article: Minkowski addition In a real vectorspace, the Minkowski sum of two (nonempty) sets S_{1} and S_{2} is defined to be the set S_{1} + S_{2} formed by the addition of vectors elementwise from the summandsets
 S_{1} + S_{2} = { x_{1} + x_{2} : x_{1} ∈ S_{1} and x_{2} ∈ S_{2} }.
More generally, the Minkowski sum of a finite family of (nonempty) sets S_{n} is the set formed by elementwise addition of vectors
 ∑ S_{n} = { ∑ x_{n} : x_{n} ∈ S_{n} }.
For Minkowski addition, the zero set {0} containing only the zero vector 0 has special importance: For every nonempty subset S of a vector space
 S + {0} = S;
in algebraic terminology, the zero vector 0 is the identity element of Minkowski addition (on the collection of nonempty sets).^{[3]}
Convex hulls of Minkowski sums
Minkowski addition behaves well with respect to the operation of taking convex hulls, as shown by the following proposition:
 For all subsets S_{1} and S_{2} of a real vectorspace, the convex hull of their Minkowski sum is the Minkowski sum of their convex hulls
 Conv( S_{1} + S_{2} ) = Conv( S_{1} ) + Conv( S_{2} ).
This result holds more generally for each finite collection of nonempty sets
 Conv( ∑ S_{n} ) = ∑ Conv( S_{n} ).
In mathematical terminology, the operations of Minkowski summation and of forming convex hulls are commuting operations.^{[4]}^{[5]}
Closed convex sets
Closed convex sets can be characterised as the intersections of closed halfspaces (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 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 halfspace H that contains C and not P. The supporting hyperplane theorem is a special case of the Hahn–Banach theorem of functional analysis.
The Minkowski sum of two compact convex sets is closed, as is the sum of a compact convex set and a closed convex set.^{[6]}
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.
Starconvex sets
Main article: Star domainLet C be a set in a real or complex vector space. C is star convex if there exists an x_{0} in C such that the line segment from x_{0} to any point y in C is contained in C. Hence a nonempty convex set is always starconvex but a starconvex set is not always convex.
Orthogonal convexity
Main article: Orthogonal convex hullAn example of generalized convexity is orthogonal convexity.^{[7]}
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.
Not 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 geodesics joining any two points in the set.
Order topology
Convexity can be extended for a space X endowed with the order topology, using the total order < of the space.^{[8]}
Let . The subspace Y is a convex set if for each pair of points such that a < b, the interval is contained in Y. That is, Y is convex if and only if .
Convexity spaces
The notion of convexity may be generalised to other objects, if certain properties of convexity are selected as axioms.
Given a set X, a convexity over X is a collection of subsets of X satisfying the following axioms:^{[1]}^{[2]}
 The empty set and X 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 (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, see the convex geometries associated with antimatroids.
See also
 Convex function
 Holomorphically convex hull
 Pseudoconvexity
 Convex metric space
 Concave set
 Helly's theorem
 Carathéodory's theorem (convex hull)
 Choquet theory
 Shapley–Folkman lemma
References
 ^ ^{a} ^{b} Soltan, Valeriu, Introduction to the Axiomatic Theory of Convexity, Ştiinţa, Chişinău, 1984 (in Russian).
 ^ ^{a} ^{b} Singer, Ivan (1997). Abstract convex analysis. Canadian Mathematical Society series of monographs and advanced texts. New York: John Wiley & Sons, Inc.. pp. xxii+491. ISBN 0471160156. MR1461544.
 ^ 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
 S+∅ = ∅.
 ^ 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 (2), Second series 41: pp. 556–583. doi:10.2307/1968735. JSTOR 1968735.
 ^ 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): Schneider, Rolf (1993). Convex bodies: The Brunn–Minkowski theory. Encyclopedia of mathematics and its applications. 44. Cambridge: Cambridge University Press. pp. xiv+490. ISBN 0521352207. MR1216521.
 ^ R. T. Rockafellar, Convex analysis, Princeton University Press, Princeton, NJ, 1970. Reprint: 1997.
 ^ Rawlins G.J.E. and Wood D, "Orthoconvexity and its generalizations", in: Computational Morphology, 137152. Elsevier, 1988.
 ^ Munkres, James; Topology, Prentice Hall; 2nd edition (December 28, 1999). ISBN 0131816292.
External links
 Lectures on Convex Sets, notes by Niels Lauritzen, at Aarhus University, March 2010.
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