# Helly-Type Theorems in Property Testing

```@inproceedings{Chakraborty2014HellyTypeTI,
title={Helly-Type Theorems in Property Testing},
author={Sourav Chakraborty and Rameshwar Pratap and Sasanka Roy and Shubhangi Saraf},
booktitle={LATIN},
year={2014}
}```
• Published in LATIN 31 July 2013
• Mathematics, Computer Science
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…
2 Citations

### Colorful Helly Theorem for Piercing Boxes with Two Points

• Mathematics
ArXiv
• 2022
. Let h ( d, 2) denotes the smallest integer such that any ﬁnite collection of axis parallel boxes in R d is two-pierceable if and only if every h ( d, 2) many boxes from the collection is

### A Stepping-Up Lemma for Topological Set Systems

• Mathematics
SoCG
• 2021
The stair convexity of Bukh, Matoušek and Nivasch is used to recast a simplicial complex as a homological minor of a cubical complex to prove that a similar phenomenon holds for any topological set system F in R.

## References

SHOWING 1-10 OF 24 REFERENCES

### Intersection properties of boxes in Rd

• Mathematics
Comb.
• 1982
A family of sets is calledn-pierceable if there exists a set ofn points such that each member of the family contains at least one of the points. Helly’s theorem on intersections of convex sets

### Property Testing with Geometric Queries

• Computer Science, Mathematics
ESA
• 2001
A number of models are discussed that in the author's opinion fit best geometric problems and apply them to study geometric properties for three very fundamental and representative problems in the area: testing convex position, testing map labeling, and testing clusterability.

### Combinatorial property testing (a survey)

• Oded Goldreich
• Mathematics
Randomization Methods in Algorithm Design
• 1997
This work considers the question of determining whether a given object has a predetermined property or is \far" from any object having the property, and focuses on combinatorial properties, and speciically on graph properties.

### Property Testing: A Learning Theory Perspective

Property testing [15,9] is the study of the following class of problems: Given the ability to perform local queries concerning a particular object, the problem is to determine whether the object has a predetermined global property, or differs significantly from any object that has the property.

### Intersection patterns of convex sets

LetK1,…Kn be convex sets inRd. For 0≦i<n denote byfithe number of subsetsS of {1,2,…,n} of cardinalityi+1 that satisfy ∩{Ki∶i∈S}≠Ø. We prove:Theorem.If fd+r=0 for somer r>=0, then {fx161-1} This

### A Problem of Geometry in R n

• Mathematics
• 1979
AssTRAcr. Let 6f be a finite family of at least n + 1 convex sets in the n-dimensional Eucidean space Rn. Helly's theorem asserts that if all the (n + l)-subfamilies of IF have nonempty intersection,

### On a conjecture of Danzer and Grünbaum

• Mathematics
• 1996
The main result of the paper is that if A is a family of homothetic triangles in the plane such that any 9 of them can be pierced by two points, then all members of A can be pierced by two points.

### Property testing and its connection to learning and approximation

• Computer Science
Proceedings of 37th Conference on Foundations of Computer Science
• 1996
The authors study the question of determining whether an unknown function has a particular property or is /spl epsiv/-far from any function with that property, and devise algorithms to test whether a graph has properties such as being k-colorable or having a /spl rho/-clique.