Continuous quantitative Helly-type results

Vidal, Tomás; Galicer, Daniel; Merzbacher, Mariano

doi:10.48550/arxiv.2006.09472

Cited by 2 publications

(3 citation statements)

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“…Naszódi [27] improved the guarantee of the volume in the intersection to d −2d , mostly settling this volumetric variant. His approach, based on sparsification of John decompositions of the identity, has been improved in several articles [11,12,13,19]. A constellation of related results that adjust the function measuring the size of the intersection, the cardinality of the subfamily intersection, or the guarantee in the conclusion have since been proven [15,29,30,32].…”

Section: Introductionmentioning

confidence: 99%

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A mélange of diameter Helly-type theorems

Dillon¹,

Soberón²

2020

Preprint

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A Helly-type theorem for diameter provides a bound on the diameter of the intersection of a finite family of convex sets in R d given some information on the diameter of the intersection of all sufficiently small subfamilies. We prove fractional and colorful versions of a longstanding conjecture by Bárány, Katchalski, and Pach. We also show that a Minkowski norm admits an exact Helly-type theorem for diameter if and only if its unit ball is a polytope and prove a colorful version for those that do. Finally, we prove Helly-type theorems for the property of "containing k colinear integer points."

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Section: Introductionmentioning

confidence: 99%

“…If we relax Conjecture 1.1 to checking subfamilies of quadratic cardinality in the dimension, then an application of Nászodi's method guarantees that the diameter of F is at least d −1 (see, e.g., [19,Theorem 1.4]). To obtain a bound of d −1/2 , however, the method would require that each set be centrally symmetric.…”

Section: Introductionmentioning

confidence: 99%

A mélange of diameter Helly-type theorems

Dillon¹,

Soberón²

2020

Preprint

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“…In many recent results regarding volumetric Helly-type theorems, analytic properties of ellipsoids are key ingredients of the proofs [Nas16,Bra16,Bra17,Bra18,FVGM20,DFN19]. Results on the sparsification of John decompositions of the identity can be translated to Helly-type theorems.…”

Section: Introductionmentioning

confidence: 99%

Isometric and affine copies of a set in volumetric Helly results

Messina¹,

Soberón²

2020

Preprint

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We show that for any compact convex set K in R d and any finite family F of convex sets in R d , if the intersection of every sufficiently small subfamily of F contains an isometric copy of K of volume 1, then the intersection of the whole family contains an isometric copy of K scaled by a factor of (1−ε), where ε is positive and fixed in advance. Unless K is very similar to a disk, the shrinking factor is unavoidable. We prove similar results for affine copies of K. We show how our results imply the existence of randomized algorithms that approximate the largest copy of K that fits inside a given polytope P whose expected runtime is linear on the number of facets of P .

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Continuous quantitative Helly-type results

Cited by 2 publications

References 0 publications

A mélange of diameter Helly-type theorems

A mélange of diameter Helly-type theorems

Isometric and affine copies of a set in volumetric Helly results

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