Convex
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In mathematics, 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. Template:Wiktionary
The concepts of convexity and concavity are important in various fields, and in various fields the adjective "convex" has their own specific meanings.
- In optics; see convex lens and concave lens.
- In mathematical analysis, see convex function.
Convex 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
- (1 − t) x + t y
is in C. In other words, every point on the line segment connecting x and y is in C.
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 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.
Properties of convex sets
If <math>S<math> is a convex set, for any <math>u_1,u_2,\ldots,u_r<math> in <math>S<math>, and any non negative numbers <math>\lambda_1,\lambda_2,\ldots,\lambda_r<math> such that <math>\lambda_1+\lambda_2+\cdots+\lambda_r=1<math>, then the vector <math>\sum_{k=1}^r\lambda_k u_k<math> is in <math>S<math>.
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.
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.
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.de:Konvex fr:Convexe nl:Convex pl:Zbiór wypukły sv:Konvex