Carathéodory’s theorem

2021 ◽  
pp. 9-12
Keyword(s):  
Carathéodory’S Theorem

Carathéodory's Theorem

1972 ◽  
Vol 15 (2) ◽  
pp. 293-293 ◽  
Author(s):  
W. D. Cook ◽  
R. J. Webster
Keyword(s):  
Carathéodory’S Theorem ◽  
Simple Generalization

Despite the abundance of generalizations of Carathéodory's theorem occurring in the literature (see [1]), the following simple generalization involving infinite convex combinations seems to have passed unnoticed. Boldface letters denote points of Rn and Greek letters denote scalars.

scholarly journals On Nonsmooth Semi-Infinite Minimax Programming Problem with(Φ,ρ)-Invexity

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
X. L. Liu ◽  
G. M. Lai ◽  
C. Q. Xu ◽  
D. H. Yuan
Keyword(s):  
Optimality Conditions ◽  
Programming Problem ◽  
Necessary Optimality Conditions ◽  
Duality Theorems ◽  
Converse Duality ◽  
Minimax Programming ◽  
Carathéodory’S Theorem

We are interested in a nonsmooth minimax programming Problem (SIP). Firstly, we establish the necessary optimality conditions theorems for Problem (SIP) when using the well-known Caratheodory's theorem. Under the Lipschitz(Φ,ρ)-invexity assumptions, we derive the sufficiency of the necessary optimality conditions for the same problem. We also formulate dual and establish weak, strong, and strict converse duality theorems for Problem (SIP) and its dual. These results extend several known results to a wider class of problems.

scholarly journals Hypercompositional Algebra, Computer Science and Geometry

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1338 ◽  
Author(s):  
Gerasimos Massouros ◽  
Christos Massouros
Keyword(s):  
Computer Science ◽  
Formal Languages ◽  
Helly’S Theorem ◽  
Projective Geometries ◽  
Carathéodory’S Theorem ◽  
Close Relationship ◽  
Radon’S Theorem ◽  
Steinitz’S Theorem ◽  
Stone’S Theorem

The various branches of Mathematics are not separated between themselves. On the contrary, they interact and extend into each other’s sometimes seemingly different and unrelated areas and help them advance. In this sense, the Hypercompositional Algebra’s path has crossed, among others, with the paths of the theory of Formal Languages, Automata and Geometry. This paper presents the course of development from the hypergroup, as it was initially defined in 1934 by F. Marty to the hypergroups which are endowed with more axioms and allow the proof of Theorems and Propositions that generalize Kleen’s Theorem, determine the order and the grade of the states of an automaton, minimize it and describe its operation. The same hypergroups lie underneath Geometry and they produce results which give as Corollaries well known named Theorems in Geometry, like Helly’s Theorem, Kakutani’s Lemma, Stone’s Theorem, Radon’s Theorem, Caratheodory’s Theorem and Steinitz’s Theorem. This paper also highlights the close relationship between the hyperfields and the hypermodules to geometries, like projective geometries and spherical geometries.

Families of homeomorphisms, relative metrics, and caratheodory's theorem

1972 ◽  
Vol 13 (2) ◽  
pp. 254-265
Author(s):  
V. P. Luferenko ◽  
G. D. Suvorov
Keyword(s):  
Relative Metrics ◽  
Carathéodory’S Theorem
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