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Brouwer fixed-point theorem

Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function mapping a compact convex set to itself there is a

Minkowski addition

In geometry, the Minkowski sum (also known as dilation) of two sets of position vectors A and B in Euclidean space is formed by adding each vector in A to each vector in B, i.e., the set Analogously,

Hyperplane separation theorem

In geometry, the hyperplane separation theorem is a theorem about disjoint convex sets in n-dimensional Euclidean space. There are several rather similar versions. In one version of the theorem, if bo

Kakutani's theorem (geometry)

Kakutani's theorem is a result in geometry named after Shizuo Kakutani. It states that every convex body in 3-dimensional space has a circumscribed cube, i.e. a cube all of whose faces touch the body.

Kakutani fixed-point theorem

In mathematical analysis, the Kakutani fixed-point theorem is a fixed-point theorem for set-valued functions. It provides sufficient conditions for a set-valued function defined on a convex, compact s

Kirchberger's theorem

Kirchberger's theorem is a theorem in discrete geometry, on linear separability. The two-dimensional version of the theorem states that, if a finite set of red and blue points in the Euclidean plane h

Krein–Milman theorem

In the mathematical theory of functional analysis, the Krein–Milman theorem is a proposition about compact convex sets in locally convex topological vector spaces (TVSs). Krein–Milman theorem — A comp

Tverberg's theorem

In discrete geometry, Tverberg's theorem, first stated by Helge Tverberg, is the result that sufficiently many points in d-dimensional Euclidean space can be partitioned into subsets with intersecting

Carathéodory's theorem (convex hull)

Carathéodory's theorem is a theorem in convex geometry. It states that if a point x of Rd lies in the convex hull of a set P, then x can be written as the convex combination of at most d + 1 points in

Helly's theorem

Helly's theorem is a basic result in discrete geometry on the intersection of convex sets. It was discovered by Eduard Helly in 1913, but not published by him until 1923, by which time alternative pro

Gordan's lemma

Gordan's lemma is a lemma in convex geometry and algebraic geometry. It can be stated in several ways.
* Let be a matrix of integers. Let be the set of non-negative integer solutions of . Then there

Hadwiger's theorem

In integral geometry (otherwise called geometric probability theory), Hadwiger's theorem characterises the valuations on convex bodies in It was proved by Hugo Hadwiger.

Radon's theorem

In geometry, Radon's theorem on convex sets, published by Johann Radon in 1921, states that any set of d + 2 points in Rd can be partitioned into two sets whose convex hulls intersect. A point in the

Brunn–Minkowski theorem

In mathematics, the Brunn–Minkowski theorem (or Brunn–Minkowski inequality) is an inequality relating the volumes (or more generally Lebesgue measures) of compact subsets of Euclidean space. The origi

M. Riesz extension theorem

The M. Riesz extension theorem is a theorem in mathematics, proved by Marcel Riesz during his study of the problem of moments.

Busemann's theorem

In mathematics, Busemann's theorem is a theorem in Euclidean geometry and geometric tomography. It was first proved by Herbert Busemann in 1949 and was motivated by his theory of area in Finsler space

Alexandrov's uniqueness theorem

The Alexandrov uniqueness theorem is a rigidity theorem in mathematics, describing three-dimensional convex polyhedra in terms of the distances between points on their surfaces. It implies that convex

Cauchy's theorem (geometry)

Cauchy's theorem is a theorem in geometry, named after Augustin Cauchy. It states that convex polytopes in three dimensions with congruent corresponding faces must be congruent to each other. That is,

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