Quantitative Tverberg Theorems Over Lattices and Other Discrete Sets

Loera, Jesús A. De; Haye, R. N. La; Rolnick, David; Soberón, Pablo

doi:10.1007/s00454-016-9858-3

Cited by 13 publications

(15 citation statements)

References 39 publications

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“…Thus, it is natural to determine which results can be extended in this quantitative framework. For the volume, several advances have been made in this direction [Nas15,DLLHRS15,Sob15]. This includes optimising the original result by Bárány, Katchalski and Pach, and finding colourful versions, fractional versions and (p, q) type theorems.…”

Section: Introductionmentioning

confidence: 99%

Helly-type theorems for the diameter

Soberón

2016

Bull. London Math. Soc.

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We study versions of Helly's theorem that guarantee that the intersection of a family of convex sets in R d has a large diameter. This includes colourful, fractional and (p, q) versions of Helly's theorem. In particular, the fractional and (p, q) versions work with conditions where the corresponding Helly theorem does not. We also include variants of Tverberg's theorem, Bárány's point selection theorem and the existence of weak epsilon-nets for convex sets with diameter estimates.

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Section: Introductionmentioning

confidence: 99%

Helly-type theorems for the diameter

Soberón

2016

Bull. London Math. Soc.

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“…1 It was also shown in [16] that it follows from the work of Alon et al [2] that weak ε-nets [1] of size c(ε, r ) also exist and a ( p, q)-theorem [3] also holds, so understanding these parameters better might lead to improved ε-net bounds. It remains an interesting challenge and a popular topic to find new connections among such theorems; for some recent papers studying the Radon numbers or Tverberg theorems of various convexity spaces, see [9][10][11]14,20,23,24,27], while for a comprehensive survey, see Bárány and Soberón [6].…”

Section: Introductionmentioning

confidence: 99%

Radon Numbers Grow Linearly

Pálvölgyi

2021

Discrete Comput Geom

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Define the k-th Radon number $$r_k$$ r k of a convexity space as the smallest number (if it exists) for which any set of $$r_k$$ r k points can be partitioned into k parts whose convex hulls intersect. Combining the recent abstract fractional Helly theorem of Holmsen and Lee with earlier methods of Bukh, we prove that $$r_k$$ r k grows linearly, i.e., $$r_k\le c(r_2)\cdot k$$ r k ≤ c ( r 2 ) · k .

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“…* It was also shown in [15] that it follows from the work of Alon et al [2] that weak ε-nets [1] of size c(ε, r) also exist and a (p, q)-theorem [3] also holds, so understanding these parameters better might lead to improved ε-net bounds. It remains an interesting challenge and a popular topic to find new connections among such theorems; for some recent papers studying the Radon numbers or Tverberg theorems of various convexity spaces, see [8,9,10,13,18,21,22,24], while for a comprehensive survey, see Bárány and Soberón [5].…”

Section: Introductionmentioning

confidence: 99%