Journal ArticleDOI
Helly"s theorem with volumes
TLDR
The Steinitz lemma and Helly's theorem were proved by Pach's number as discussed by the authors, which is the number of the number used by Steinitz and Pach in their paper.Abstract:
Keywords: Steinitz lemma ; Helly's theorem Note: Professor Pach's number: [016] Reference DCG-ARTICLE-1984-002doi:10.2307/2322144 Record created on 2008-11-14, modified on 2017-05-12read more
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Book
Combinatorial Geometry
János Pach,K P Agarwal +1 more
TL;DR: This discipline emerged from number theory after the fruitful observation made by Minkowski (1896) that many important results in diophantine approximation (and in some other central fields of number theory) can be established by easy geometric arguments.
Book ChapterDOI
Helly, Radon, and Carathéodory Type Theorems
TL;DR: In this paper, the authors discuss applications and generalizations of the classical theorems of Helly, Radon, and Caratheodory, as well as their ramifications in the context of combinatorial convexity theory.
Book ChapterDOI
Geometric Transversal Theory
TL;DR: Theorem 1.1 (Helly's Theorem) of transversal theory has its origins in Helly's theorem as mentioned in this paper, which states that if every d + 1 members of a convex set have a common point, then there is a point common to all the members of the set.
Journal ArticleDOI
Proof of a Conjecture of Bárány, Katchalski and Pach
TL;DR: Barany, Katchalski and Pach as mentioned in this paper proved the following quantitative form of Helly's theorem: if the intersection of a family of convex sets in Ω(R) is of volume one, then the intersection intersection of some subfamily of at most 2d members is of a volume at most some constant v(d).
Posted Content
A Quantitative Doignon-Bell-Scarf Theorem
TL;DR: It is proved that there exists a constant c(n,k), depending only on the dimension n and k, such that if a polyhedron {x∈Rn: Ax≤b} contains exactly k integer points, thenthere exists a subset of the rows, of cardinality no more than c( n,k,), defining apolyhedron that contains exactly the same kinteger points.
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