Look up **Convex set** in Wiktionary, the free dictionary. *For other uses of convex, see convex function and convexity.* In Euclidean space, an object is **convex** if for any pair of points within the object, any 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 is not convex. Wikipedia does not have an article with this exact name. ...
Wiktionary is a Wikimedia Foundation project intended to be a free wiki dictionary (hence: Wiktionary) (including thesaurus and lexicon) in every language. ...
In mathematics, convex function is a real-valued function f defined on an interval (or on any convex subset C of some vector space), if for any two points x and y in its domain C and any t in [0,1], we have Convex function on an interval. ...
Convexity may refer to one of the following. ...
In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...
A line, or straight line, is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object, i. ...
Three dimensions A cube (or hexahedron) is a Platonic solid composed of six square faces, with three meeting at each vertex. ...
## Convex sets
A non-convex (concave) set. Let *C* be a set in a real or complex vector space. *C* is said to be **convex** if, for all *x* and *y* in *C* and all *t* in the interval [0,1], the point Image File history File links Convex_polygon_illustration1. ...
Image File history File links Convex_polygon_illustration1. ...
Image File history File links Convex_polygon_illustration2. ...
Image File history File links Convex_polygon_illustration2. ...
In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ...
In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite lineâ€”the number line. ...
Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = âˆ’1. ...
Vector spaces (or linear spaces) are spaces whose elements, known as vectors, can be scaled and added; all linear combinations can be formed. ...
In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ...
- (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 is connected. In mathematics, a line segment is a part of a line that is bounded by two end points. ...
Connected and disconnected subspaces of RÂ². The space A at top is connected; the shaded space B at bottom is not. ...
A set *C* is called **absolutely convex** if it is convex and balanced. In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field K with an absolute value |.|) is a set S so that for all scalars Î± with |Î±| â‰¤ 1 with The balanced hull or balanced envelope for a set S is...
The convex subsets of **R** (the set of real numbers) are simply the intervals of **R**. Some examples of convex subsets of Euclidean 2-space are regular polygons and bodies of constant width. Some examples of convex subsets of Euclidean 3-space are the Archimedean solids and the Platonic solids. The Kepler-Poinsot solids are examples of non-convex sets. A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. The relationship of one set being a subset of another is called inclusion. ...
In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...
A polygon (from the Greek poly, for many, and gonos, for angle) is a closed planar path composed of a finite number of sequential straight line segments. ...
For closed convex planar bodies whose boundary is a smooth curve, one notes that there are exactly two parallel tangent lines to the boundary curve in any given direction. ...
In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...
In geometry an Archimedean solid or semi-regular solid is a semi-regular convex polyhedron composed of two or more types of regular polygon meeting in identical vertices. ...
In solid geometry and some ancient physical theories, a Platonic solid is a convex polyhedron with: All its faces being congruent regular polygons The same number of faces meeting at each of its vertices These are in contrast to: The Kepler-Poinsot solids, which are not convex The Archimedean and...
A Kepler solid (also called Kepler-Poinsot solid) is a regular non-convex polyhedron, all the faces of which are identical regular polygons and which has the same number of faces meeting at all its vertices (compare to Platonic solids). ...
### Properties of convex sets If *S* is a convex set, for any in *S*, and any non negative numbers such that , then the vector is in *S*. A negative number is a number that is less than zero, such as −3. ...
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. This also means that any 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 name lattice is suggested by the form of the Hasse diagram depicting it. ...
Convex Hull: Elastic band analogy // Alternative definitions 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. (Note that X may be the union of any set of objects made of points). ...
Closed convex sets can be characterised as the intersections of *closed half-spaces* (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 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. In topology and related branches of mathematics, a closed set is a set whose complement is open. ...
In geometry, a half-space is any of the two parts into which a hyperplane divides an affine space. ...
A hyperplane is a concept in geometry. ...
In mathematics, the Hahn-Banach theorem is a central tool in functional analysis. ...
Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ...
## Star-convex sets Let *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 convex set is always star convex but a star-convex object is not always convex.
## Non-Euclidean geometry The definition of a convex set and a convex hull extends naturally to non-Euclidean geometry by defining a convex set to contain the geodesics joining any two points in the set. Behavior of lines with a common perpendicular in each of the three types of geometry The term non-Euclidean geometry (also spelled: non-Euclidian geometry) describes both hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry. ...
In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. Definition of geodesic depends on the type of curved space. If the space carries a natural metric then geodesics are defined to be (locally) the shortest path between points on the space. ...
## Generalized 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.
## Abstract (axiomatic) convexity The notion of convexity may be generalised to other objects, if certain properties of convexity are selected as axioms. Convexity may refer to one of the following. ...
An axiom is a sentence or proposition that is taken for granted as true, and serves as a starting point for deducing other truths. ...
Given a set *X*, the **convexity** over *X* is a subset of powerset of *X* that satisfies the following axioms. In mathematics, given a set S, the power set of S, written P(S) or 2S, is the set of all subsets of S. In formal language, the existence of power set of any set is presupposed by the axiom of power set. ...
- 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 the **convexity space**. For the ordinary convexity, the first two axioms hold, and the third one is trivial. In mathematics, a total order, linear order or simple order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ...
## See also In mathematics, more precisely in the theory of functions of several complex variables, a pseudoconvex set is a special type of open set in the n-dimensional complex space Cn. ...
## References - Rawlins G.J.E. and Wood D, "Ortho-convexity and its generalizations", in:
*Computational Morphology*, 137-152. Elsevier, 1988. - Soltan, Valeriu,
*Introduction to the Axiomatic Theory of Convexity*, Ştiinţa, Chişinău, 1984 (in Russian). |