The Geometry of Radon's Theorem

1972 ◽  
Vol 79 (9) ◽  
pp. 949 ◽  
Author(s):  
B. B. Peterson
Keyword(s):  
Radon’S Theorem

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.

The Geometry of Radon's Theorem

1972 ◽  
Vol 79 (9) ◽  
pp. 949-963 ◽  
Author(s):  
B. B. Peterson
Keyword(s):  
Radon’S Theorem

Neighbourliness and Radon's theorem

Mathematika ◽  
1969 ◽  
Vol 16 (2) ◽  
pp. 273-275 ◽  
Author(s):  
G. C. Shephard
Keyword(s):  
Radon’S Theorem

scholarly journals An extension of Radon's theorem

1968 ◽  
Vol 12 (2) ◽  
pp. 184-189 ◽  
Author(s):  
John R. Reay
Keyword(s):  
Radon’S Theorem

scholarly journals A generalization of Radon's theorem II

1981 ◽  
Vol 24 (3) ◽  
pp. 321-325 ◽  
Author(s):  
H. Tverberg
Keyword(s):  
Dimensional Space ◽  
Convex Polytopes ◽  
Empty Intersection ◽  
Radon’S Theorem ◽  
Positive Integers

A new proof is given of the following result: Let m and d be positive integers, and let a set of md + m − d points be given in d-dimensional space. Then the set can be partitioned into m sets such that the m convex polytopes spanned by the sets have a non-empty intersection.

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