Helly’s Theorem and Strongly Proximal Helly Theorem

Helly's Theorem with Volumes

American Mathematical Monthly ◽

10.2307/2322144 ◽

1984 ◽

Vol 91 (6) ◽

pp. 362 ◽

Cited By ~ 8

Author(s):

I. Barany ◽

M. Katchalski ◽

Janos Pach

Keyword(s):

Helly’S Theorem

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Helly’s theorem: New variations and applications

Algebraic and Geometric Methods in Discrete Mathematics - Contemporary Mathematics ◽

10.1090/conm/685/13718 ◽

2017 ◽

pp. 55-95 ◽

Cited By ~ 3

Author(s):

Nina Amenta ◽

Jesús De Loera ◽

Pablo Soberón

Keyword(s):

Helly’S Theorem

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Helly’s theorem

10.1090/ulect/077/07 ◽

2021 ◽

pp. 29-32

Keyword(s):

Helly’S Theorem

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Helly-Type Theorems in Property Testing

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195918500115 ◽

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.

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