Helly's Theorem with Volumes

1984 ◽  
Vol 91 (6) ◽  
pp. 362 ◽  
Author(s):  
I. Barany ◽  
M. Katchalski ◽  
Janos Pach
Keyword(s):  
Helly’S Theorem

scholarly journals Helly’s theorem: New variations and applications

2017 ◽  
pp. 55-95 ◽  
Author(s):  
Nina Amenta ◽  
Jesús De Loera ◽  
Pablo Soberón
Keyword(s):  
Helly’S Theorem

Helly’s theorem

2021 ◽  
pp. 29-32
Keyword(s):  
Helly’S Theorem

scholarly journals Helly-Type Theorems in Property Testing

2018 ◽  
Vol 28 (04) ◽  
pp. 365-379
Author(s):  
Sourav Chakraborty ◽  
Rameshwar Pratap ◽  
Sasanka Roy ◽  
Shubhangi Saraf
Keyword(s):  
High Probability ◽  
Convex Sets ◽  
Property Testing ◽  
Geometric Object ◽  
Fundamental Result ◽  
Symmetric Convex ◽  
Helly’S Theorem ◽  
Convex Object ◽  
Constant Size ◽  
Testing Algorithm

Helly’s theorem is a fundamental result in discrete geometry, describing the ways in which convex sets intersect with each other. If [Formula: see text] is a set of [Formula: see text] points in [Formula: see text], we say that [Formula: see text] is [Formula: see text]-clusterable if it can be partitioned into [Formula: see text] clusters (subsets) such that each cluster can be contained in a translated copy of a geometric object [Formula: see text]. In this paper, as an application of Helly’s theorem, by taking a constant size sample from [Formula: see text], we present a testing algorithm for [Formula: see text]-clustering, i.e., to distinguish between the following two cases: when [Formula: see text] is [Formula: see text]-clusterable, and when it is [Formula: see text]-far from being [Formula: see text]-clusterable. A set [Formula: see text] is [Formula: see text]-far [Formula: see text] from being [Formula: see text]-clusterable if at least [Formula: see text] points need to be removed from [Formula: see text] in order to make it [Formula: see text]-clusterable. We solve this problem when [Formula: see text], and [Formula: see text] is a symmetric convex object. For [Formula: see text], we solve a weaker version of this problem. Finally, as an application of our testing result, in the case of clustering with outliers, we show that with high probability one can find the approximate clusters by querying only a constant size sample.

Helly's Theorem with Volumes

1984 ◽  
Vol 91 (6) ◽  
pp. 362-365 ◽  
Author(s):  
I. Bárány ◽  
M. Katchalski ◽  
János Pach
Keyword(s):  
Helly’S Theorem

A Proof of the Equivalence of Helly's and Krasnoselski's Theorems

1977 ◽  
Vol 20 (1) ◽  
pp. 35-37 ◽  
Author(s):  
J. Borwein
Keyword(s):  
Helly’S Theorem

AbstractWe show that Krasnoselski's Theorem, which is usually derived from Helly's Theorem, is in fact equivalent to it.

scholarly journals Group actions and Helly's theorem

2009 ◽  
Vol 222 (5) ◽  
pp. 1574-1588 ◽  
Author(s):  
Benson Farb
Keyword(s):  
Group Actions ◽  
Helly’S Theorem

scholarly journals On the semi-sub-hypergroups of a hypergroup

1991 ◽  
Vol 14 (2) ◽  
pp. 293-304 ◽  
Author(s):  
Ch. G. Massouros
Keyword(s):  
General Theorem ◽  
Helly’S Theorem

In this paper we study some properties of the semi-sub-hypergroups and the closed sub-hypergroups of the hypergroups. We introduce the correlated elements and the fundamental elements and we connect the concept antipodal of the latter with Frattin's hypergroup. We also present Helly's Theorem as a corollary of a more general Theorem.

A Variant of Helly's Theorem

1960 ◽  
Vol 11 (4) ◽  
pp. 517 ◽  
Author(s):  
Branko Grunbaum
Keyword(s):  
Helly’S Theorem

scholarly journals Eckhoff's Problem on Convex Sets in the Plane

10.37236/9978 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Adam S. Jobson ◽  
André E. Kézdy ◽  
Jenő Lehel
Keyword(s):  
Convex Sets ◽  
Common Point ◽  
Finite Family ◽  
Helly’S Theorem ◽  
Uniform Hypergraphs ◽  
Special Cases

Eckhoff proposed a combinatorial version of the classical Hadwiger–Debrunner $(p,q)$-problems as follows. Let ${\cal F}$ be a finite family of convex sets in the plane and  let $m\geqslant 1$ be an integer. If among every ${m+2\choose 2}$ members of ${\cal F}$ all but at most $m-1$ members have a common point, then there is a common point for all but at most $m-1$ members of ${\cal F}$. The claim is an extension of Helly's theorem ($m=1$). The case $m=2$ was verified by Nadler and by Perles. Here we show that Eckhoff 's conjecture follows from an old conjecture due to Szemerédi and Petruska concerning $3$-uniform hypergraphs. This conjecture is still open in general; its  solution for a few special cases answers Eckhoff's problem for $m=3,4$. A new proof for the case $m=2$ is also presented.

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