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Table of Contents

1. 22 relations: American Mathematical Society, Choquet theory, Compact space, Conical combination, Constantin Carathéodory, Convex combination, Convex geometry, Convex hull, Ernst Steinitz, Helly's theorem, Kirchberger's theorem, Krein–Milman theorem, N-dimensional polyhedron, Ordered field, Perron–Frobenius theorem, PlanetMath, PLS (complexity), PPAD (complexity), Radon's theorem, Shapley–Folkman lemma, Simplex, Tverberg's theorem.

2. Convex hulls
3. Geometric transversal theory
4. Theorems in convex geometry
5. Theorems in discrete geometry

American Mathematical Society

The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs.

See Carathéodory's theorem (convex hull) and American Mathematical Society

Choquet theory

In mathematics, Choquet theory, named after Gustave Choquet, is an area of functional analysis and convex analysis concerned with measures which have support on the extreme points of a convex set C. Roughly speaking, every vector of C should appear as a weighted average of extreme points, a concept made more precise by generalizing the notion of weighted average from a convex combination to an integral taken over the set E of extreme points. Carathéodory's theorem (convex hull) and Choquet theory are convex hulls.

See Carathéodory's theorem (convex hull) and Choquet theory

Compact space

In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space.

See Carathéodory's theorem (convex hull) and Compact space

Conical combination

Given a finite number of vectors x_1, x_2, \dots, x_n in a real vector space, a conical combination, conical sum, or weighted sumConvex Analysis and Minimization Algorithms by Jean-Baptiste Hiriart-Urruty, Claude Lemaréchal, 1993,, Mathematical Programming, by Melvyn W. Jeter (1986), of these vectors is a vector of the form where \alpha_i are non-negative real numbers.

See Carathéodory's theorem (convex hull) and Conical combination

Constantin Carathéodory

Constantin Carathéodory (Konstantinos Karatheodori; 13 September 1873 – 2 February 1950) was a Greek mathematician who spent most of his professional career in Germany.

See Carathéodory's theorem (convex hull) and Constantin Carathéodory

Convex combination

In convex geometry and vector algebra, a convex combination is a linear combination of points (which can be vectors, scalars, or more generally points in an affine space) where all coefficients are non-negative and sum to 1. Carathéodory's theorem (convex hull) and convex combination are convex hulls.

See Carathéodory's theorem (convex hull) and Convex combination

Convex geometry

In mathematics, convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space.

See Carathéodory's theorem (convex hull) and Convex geometry

Convex hull

In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. Carathéodory's theorem (convex hull) and convex hull are convex hulls.

See Carathéodory's theorem (convex hull) and Convex hull

Ernst Steinitz

Ernst Steinitz (13 June 1871 – 29 September 1928) was a German mathematician.

See Carathéodory's theorem (convex hull) and Ernst Steinitz

Helly's theorem

Helly's theorem is a basic result in discrete geometry on the intersection of convex sets. Carathéodory's theorem (convex hull) and Helly's theorem are geometric transversal theory, theorems in convex geometry and theorems in discrete geometry.

See Carathéodory's theorem (convex hull) and Helly's theorem

Kirchberger's theorem

Kirchberger's theorem is a theorem in discrete geometry, on linear separability. Carathéodory's theorem (convex hull) and Kirchberger's theorem are theorems in convex geometry and theorems in discrete geometry.

See Carathéodory's theorem (convex hull) and Kirchberger's theorem

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). Carathéodory's theorem (convex hull) and Krein–Milman theorem are convex hulls, theorems in convex geometry and theorems in discrete geometry.

See Carathéodory's theorem (convex hull) and Krein–Milman theorem

N-dimensional polyhedron

An n-dimensional polyhedron is a geometric object that generalizes the 3-dimensional polyhedron to an n-dimensional space.

See Carathéodory's theorem (convex hull) and N-dimensional polyhedron

Ordered field

In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations.

See Carathéodory's theorem (convex hull) and Ordered field

Perron–Frobenius theorem

In matrix theory, the Perron–Frobenius theorem, proved by and, asserts that a real square matrix with positive entries has a unique eigenvalue of largest magnitude and that eigenvalue is real.

See Carathéodory's theorem (convex hull) and Perron–Frobenius theorem

PlanetMath

PlanetMath is a free, collaborative, mathematics online encyclopedia.

See Carathéodory's theorem (convex hull) and PlanetMath

PLS (complexity)

In computational complexity theory, Polynomial Local Search (PLS) is a complexity class that models the difficulty of finding a locally optimal solution to an optimization problem.

See Carathéodory's theorem (convex hull) and PLS (complexity)

PPAD (complexity)

In computer science, PPAD ("Polynomial Parity Arguments on Directed graphs") is a complexity class introduced by Christos Papadimitriou in 1994.

See Carathéodory's theorem (convex hull) and PPAD (complexity)

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. Carathéodory's theorem (convex hull) and Radon's theorem are convex hulls, geometric transversal theory, theorems in convex geometry and theorems in discrete geometry.

See Carathéodory's theorem (convex hull) and Radon's theorem

Shapley–Folkman lemma

The Shapley–Folkman lemma is a result in convex geometry that describes the Minkowski addition of sets in a vector space. Carathéodory's theorem (convex hull) and Shapley–Folkman lemma are convex hulls and geometric transversal theory.

See Carathéodory's theorem (convex hull) and Shapley–Folkman lemma

Simplex

In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions.

See Carathéodory's theorem (convex hull) and Simplex

Tverberg's theorem

In discrete geometry, Tverberg's theorem, first stated by Helge Tverberg in 1966, is the result that sufficiently many points in d-dimensional Euclidean space can be partitioned into subsets with intersecting convex hulls. Carathéodory's theorem (convex hull) and Tverberg's theorem are convex hulls, geometric transversal theory, theorems in convex geometry and theorems in discrete geometry.

See Carathéodory's theorem (convex hull) and Tverberg's theorem

See also

Convex hulls

• 0/1-polytope
• Alpha shape
• Carathéodory's theorem (convex hull)
• Choquet theory
• Convex combination
• Convex hull
• Convex hull algorithms
• Convex hull of a simple polygon
• Convex layers
• Convex position
• Convexity in economics
• Extreme point
• Krein–Milman theorem
• Local convex hull
• Non-convexity (economics)
• Orthogonal convex hull
• Potato peeling
• Radon's theorem
• Relative convex hull
• Shapley–Folkman lemma
• Tverberg's theorem

Geometric transversal theory

• Carathéodory's theorem (convex hull)
• Helly's theorem
• Radon's theorem
• Shapley–Folkman lemma
• Tverberg's theorem

Theorems in convex geometry

• Alexandrov's uniqueness theorem
• Brouwer fixed-point theorem
• Brunn–Minkowski theorem
• Busemann's theorem
• Carathéodory's theorem (convex hull)
• Cauchy's theorem (geometry)
• Doignon's theorem
• Gordan's lemma
• Hadwiger's theorem
• Helly's theorem
• Hyperplane separation theorem
• Kakutani fixed-point theorem
• Kakutani's theorem (geometry)
• Kirchberger's theorem
• Krein–Milman theorem
• M. Riesz extension theorem
• Minkowski addition
• Radon's theorem
• Tverberg's theorem

Theorems in discrete geometry

• Alexandrov's uniqueness theorem
• Balinski's theorem
• Beck's theorem (geometry)
• Carathéodory's theorem (convex hull)
• Cauchy's theorem (geometry)
• De Bruijn's theorem
• De Bruijn–Erdős theorem (incidence geometry)
• Doignon's theorem
• Erdős–Anning theorem
• Erdős–Nagy theorem
• Erdős–Szekeres theorem
• Four-vertex theorem
• Helly's theorem
• Kirchberger's theorem
• Krein–Milman theorem
• Monsky's theorem
• Radon's theorem
• Steinitz's theorem
• Sylvester–Gallai theorem
• Szemerédi–Trotter theorem
• Tverberg's theorem
• Wallace–Bolyai–Gerwien theorem

References

[1] https://en.wikipedia.org/wiki/Carathéodory's_theorem_(convex_hull)

Also known as Carathéodory theorem (convex hull), Colorful caratheodory theorem.

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