منابع مشابه
Computational Aspects of the Colorful Carathéodory Theorem
Let P1, . . . , Pd+1 ⊂ R be d-dimensional point sets such that the convex hull of each Pi contains the origin. We call the sets Pi color classes, and we think of the points in Pi as having color i. A colorful choice is a set with at most one point of each color. The colorful Carathéodory theorem guarantees the existence of a colorful choice whose convex hull contains the origin. So far, the com...
متن کاملAn optimal generalization of the Colorful Carathéodory theorem
The Colorful Carathéodory theorem by Bárány (1982) states that given d+1 sets of points in R, the convex hull of each containing the origin, there exists a simplex (called a ‘rainbow simplex’) with at most one point from each point set, which also contains the origin. Equivalently, either there is a hyperplane separating one of these d+ 1 sets of points from the origin, or there exists a rainbo...
متن کاملA noncommutative version of the Julia-Wolff-Carathéodory theorem
The classical Julia–Wolff–Carathéodory theorem characterizes the behaviour of the derivative of an analytic self-map of a unit disk or of a half-plane of the complex plane at certain boundary points. We prove a version of this result that applies to noncommutative self-maps of noncommutative half-planes in von Neumann algebras at points of the distinguished boundary of the domain. Our result, s...
متن کاملA further generalization of the colourful Carathéodory theorem
Given d +1 sets, or colours, S1,S2, . . . ,Sd+1 of points in Rd , a colourful set is a set S ⊂⋃i Si such that |S ∩Si | ≤ 1 for i = 1, . . . ,d +1. The convex hull of a colourful set S is called a colourful simplex. Bárány’s colourful Carathéodory theorem asserts that if the origin 0 is contained in the convex hull of Si for i = 1, . . . ,d + 1, then there exists a colourful simplex containing 0...
متن کاملPolynomial Extensions of the Milliken-taylor Theorem
Milliken-Taylor systems are some of the most general infinitary configurations that are known to be partition regular. These are sets of the form MT (〈ai〉i=1, 〈xn〉n=1) = { ∑m i=1 ai ∑ t∈Fi xt : F1, F2, . . . , Fm are increasing finite nonempty subsets of N}, where a1, a2, . . . , am ∈ Z with am > 0 and 〈xn〉n=1 is a sequence in N. That is, if p(y1, y2, . . . , ym) = ∑m i=1 aiyi is a given linear...
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ژورنال
عنوان ژورنال: Proceedings of the American Mathematical Society
سال: 1968
ISSN: 0002-9939
DOI: 10.1090/s0002-9939-1968-0220914-7