Steinitz’s Theorem and B-Convexity

Hypercompositional Algebra, Computer Science and Geometry

Mathematics ◽

10.3390/math8081338 ◽

2020 ◽

Vol 8 (8) ◽

pp. 1338 ◽

Cited By ~ 1

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.

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An analogue of a theorem of Steinitz for ball polyhedra in $$\mathbb {R}^3$$

Aequationes Mathematicae ◽

10.1007/s00010-021-00815-9 ◽

2021 ◽

Author(s):

Sami Mezal Almohammad ◽

Zsolt Lángi ◽

Márton Naszódi

Keyword(s):

Convex Polyhedron ◽

3 Dimensional ◽

Edge Graph ◽

Steinitz’S Theorem

AbstractSteinitz’s theorem states that a graph G is the edge-graph of a 3-dimensional convex polyhedron if and only if, G is simple, plane and 3-connected. We prove an analogue of this theorem for ball polyhedra, that is, for intersections of finitely many unit balls in $$\mathbb {R}^3$$ R 3 .

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