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. History Main article: History of mathematics In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. ... 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. ...

The concepts of convexity and concavity are important in various fields, and in various fields the adjective "convex" has their own specific meanings. File links The following pages link to this file: Alchemy Ada Adventure Apartheid Abbreviation Airplane (disambiguation) Abduction Alder Anno Domini Air ABC (disambiguation) Ad hominem Afghan AD Aether Aba Anus Affinity Ai AZ Albinism Accumulator Binary Chess Computer Carbon Cow Cricket (disambiguation) Collection Convex Culture Ceramics Case Creation Crow (disambiguation... Wiktionary is a sister project to Wikipedia intended to be a free wiki dictionary (thesaurus, lexicon therein) in every language. ...

• In optics; see convex lens and concave lens.
• In mathematical analysis, see convex function.

See also list of optical topics. ... Convex lens converging light rays A convex lens is a lens that is curved outward (convex): the ends are narrow and the middle is wide. ... concave lens diverging light rays A concave lens is a lens with inward-curving (concave) surfaces: the ends are wide, the middle is small. ... Analysis is that branch of mathematics which deals with the real numbers, complex numbers, and their functions. ... A 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 interval I.e. ...

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 The text or formatting below is generated by a template which has been proposed for deletion. ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ... The fundamental concept in linear algebra is that of a vector space or linear space. ... In elementary algebra, an interval is a set that contains every real number between two indicated numbers, and possibly the two numbers themselves. ...

(1 − t) x + t y

is in C. In other words, every point on the line segment connecting x and y is in C. In mathematics, a line segment is a part of a line that is bounded by two end points. ...

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. In mathematics and astronomy, 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 and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... An Archimedean solid or semiregular solid is a convex polyhedron with regular polygons as faces, such that at least two different types of regular polygons are used, and all vertices are identical (in the sense that the polygons are arranged in the same way about each vertex, and if someone... A Platonic solid is a convex polyhedron whose faces all use the same regular polygon and such that the same number of faces meet at all its vertices. ... 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.

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. See lattice for other mathematical as well as non-mathematical meanings of the term. ... In mathematics, the convex hull for an object or a set of objects is the minimal convex set containing the given objects. ...

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. ... In geometry, a hyperplane is a linear, affine, or projective subspace of codimension 1. ... In mathematics, the Hahn-Banach theorem is a central tool in functional analysis. ... Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. ...

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. 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. ...