scholarly journals A Further Generalization of the Colourful Carathéodory Theorem

2013 ◽  
pp. 179-190 ◽  
Author(s):  
Frédéric Meunier ◽  
Antoine Deza
Keyword(s):  
Colourful Carathéodory Theorem ◽  
Carathéodory Theorem

Robust Tverberg and Colourful Carathéodory Results via Random Choice

2017 ◽  
Vol 27 (3) ◽  
pp. 427-440 ◽  
Author(s):  
PABLO SOBERÓN
Keyword(s):  
Probabilistic Method ◽  
Convex Hulls ◽  
Random Choice ◽  
Tverberg’S Theorem ◽  
Colourful Carathéodory Theorem ◽  
Carathéodory Theorem

We use the probabilistic method to obtain versions of the colourful Carathéodory theorem and Tverberg's theorem with tolerance.In particular, we give bounds for the smallest integer N = N(t,d,r) such that for any N points in ℝd, there is a partition of them into r parts for which the following condition holds: after removing any t points from the set, the convex hulls of what is left in each part intersect.We prove a bound N = rt + O($\sqrt{t}$) for fixed r,d which is polynomial in each parameters. Our bounds extend to colourful versions of Tverberg's theorem, as well as Reay-type variations of this theorem.

scholarly journals On the Boundary Dieudonné–Pick Lemma

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1108
Author(s):  
Olga Kudryavtseva ◽  
Aleksei Solodov
Keyword(s):  
Fixed Point ◽  
Interior Point ◽  
Sharp Estimate ◽  
General Boundary ◽  
Extremal Functions ◽  
Angular Derivative ◽  
Additional Information ◽  
Carathéodory Theorem ◽  
Class Of Functions ◽  
Boundary Fixed Point

The class of holomorphic self-maps of a disk with a boundary fixed point is studied. For this class of functions, the famous Julia–Carathéodory theorem gives a sharp estimate of the angular derivative at the boundary fixed point in terms of the image of the interior point. In the case when additional information about the value of the derivative at the interior point is known, a sharp estimate of the angular derivative at the boundary fixed point is obtained. As a consequence, the sharpness of the boundary Dieudonné–Pick lemma is established and the class of the extremal functions is identified. An unimprovable strengthening of the Osserman general boundary lemma is also obtained.

scholarly journals An optimal generalization of the Colorful Carathéodory theorem

2016 ◽  
Vol 339 (4) ◽  
pp. 1300-1305 ◽  
Author(s):  
Nabil H. Mustafa ◽  
Saurabh Ray
Keyword(s):  
Colorful Carathéodory Theorem ◽  
Carathéodory Theorem

scholarly journals An Effective Carathéodory Theorem

2011 ◽  
Vol 50 (4) ◽  
pp. 579-588 ◽  
Author(s):  
Timothy H. McNicholl
Keyword(s):  
Carathéodory Theorem

scholarly journals The Lindelöf principle and angular derivatives in convex domains of finite type

2002 ◽  
Vol 73 (2) ◽  
pp. 221-250 ◽  
Author(s):  
Marco Abate ◽  
Roberto Tauraso
Keyword(s):  
Finite Type ◽  
Boundary Behaviour ◽  
Convex Domains ◽  
Kobayashi Distance ◽  
Angular Derivatives ◽  
Circular Domains ◽  
Lindelöf Principle ◽  
Real Analytic ◽  
Bounded Convex ◽  
Carathéodory Theorem

AbstractWe describe a generalization of the classical Julia-Wolff-Carathéodory theorem to a large class of bounded convex domains of finite type, including convex circular domains and convex domains with real analytic boundary. The main tools used in the proofs are several explicit estimates on the boundary behaviour of Kobayashi distance and metric, and a new Lindelöf principle.

Carathéodory Theorem

2006 ◽  
pp. 236-237
Author(s):  
Gabriele E. Danninger-Uchida
Keyword(s):  
Carathéodory Theorem

The Structure of C*-Convex Sets

1994 ◽  
Vol 46 (5) ◽  
pp. 1007-1026 ◽  
Author(s):  
Phillip B. Morenz
Keyword(s):  
Convex Sets ◽  
Extreme Points ◽  
Structural Elements ◽  
The Right ◽  
The Relationship ◽  
Carathéodory Theorem ◽  
Convex Subsets

AbstractCompact C*-convex subsets of Mn correspond exactly to n-th matrix ranges of operators. The main result of this paper is to discover the “right” analog of linear extreme points, called structural elements, and then to prove a generalised Krein-Milman theorem for C*-convex subsets of Mn. The relationship between structural elements and an earlier attempted generalisation, called C*-extreme points, is examined, solving affirmatively a conjecture of Loebl and Paulsen [8]. An improved bound for a C* -convex version of the Caratheodory theorem for convex sets is also given.

scholarly journals Topological vitali measure spaces

1985 ◽  
Vol 32 (2) ◽  
pp. 225-249
Author(s):  
D.N. Sarkhel ◽  
T. Chakraborti
Keyword(s):  
Measure Space ◽  
Density Theorem ◽  
Main Theme ◽  
Abstract Space ◽  
Covering Theorem ◽  
Approximate Continuity ◽  
Measure Spaces ◽  
General Abstract ◽  
Carathéodory Theorem

The properties of Lebesgue outer measures embodied in the Vitali covering theorem, the Vitali-Carathéodory theorem, the Lusin theorem, the density theorem, outer regularity and inner regularity, and the relation between measurability and approximate continuity are studied in a general abstract space, called a topological Vitali measure space. The main theme is the mutual equivalence of these properties.

Sign in / Sign up

Export Citation Format

Share Document