| • Science | • People | • Locations | • Timeline |
| Contents | ||
The concept of convexity and concavity are important in optics; see convex lens and concave lens.
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
is in C. In other words, every point on the line segment connecting x and y is in C.
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 solidsA 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. Compare with the Kepler-Poinsot solids, which are not convex, and the Archimedean and Johnson solids, whi. The Kepler-Poinsot solidA 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). There are four difs are examples of non-convex sets.
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 latticeSee lattice for other mathematical as well as non-mathematical meanings of the term. In mathematics, a lattice is a partially ordered set in which all nonempty finite subsets have both a supremum join and an infimum meet . On the other hand, lattices can. This also means that any subset A of the vector space is contained within a smallest convex set (called the convex hullIn mathematics, the convex hull for an object or a set of objects is the minimal convex set containing the given objects. It is the minimal convex set because the convex hull is a subset of any convex set which contains the given objects. The convex hull of A), namely the intersection of all convex sets containing A.
ClosedIn topology and related branches of mathematics, a set is called closed if its complement is open. This implies that a closed set contains its own boundary. Intuitively, if you are outside the set, and you "wiggle" a little bit, you will still be outside convex sets can be characterised as the intersections of closed half-spaceIn geometry, a half-space is any of the two parts into which a hyperplane divides an affine space. More strictly, an open half-space is any of the two open sets produced by the subtraction of a hyperplane from the affine space. A closed half-space is thes (sets of point in space that lie on and to one side of a hyperplaneIn geometry, a hyperplane is a linear, affine, or projective subspace of codimension 1. In particular, in a three-dimensional space, a hyperplane is the usual plane. In a two-dimensional space, a hyperplane is a line. In a one-dimensional space, a hyperpl). 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 theoremFunctional analysis Theorems In mathematics, the Hahn-Banach theorem is a central tool in functional analysis. It allows one to extend linear operators defined on a subspace of some vector space to the whole space, and it also shows that there are "enough of functional analysis.