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Bounding Radon's number via Betti numbers in the plane. Patak, Pavel
Description
In her recent result, Zuzana Patáková has shown that for a finite family $\mathcal F$ of sets in $\mathbb R^d$, one can use Betti numbers of intersections of subfamilies of $\mathcal F$, to bound the Radon's number of $\mathcal F$. The result has interesting consequences, some of them are easy or standard, other follow from a result of Holmsen and Lee. Let me name just few: variants of Helly's, Tverberg's, colorful and fractional Helly theorems, existence of weak $\varepsilon$-nets, $(p,q)$-theorems, \dots Nevertheless, the original bounds on Radon's number are too large to be widely applicable. We improve the situation in the plane and show how to obtain polynomial bounds. More generally the result extends to other two-dimensional (pseudo)manifolds.
Item Metadata
Title |
Bounding Radon's number via Betti numbers in the plane.
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Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
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Date Issued |
2019-10-09T11:03
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Description |
In her recent result, Zuzana Patáková has shown that
for a finite family $\mathcal F$ of sets in $\mathbb R^d$,
one can use Betti numbers of intersections of subfamilies of $\mathcal
F$, to bound the Radon's number of $\mathcal F$.
The result has interesting consequences, some of them are easy or
standard, other follow from a result of Holmsen and Lee. Let me name
just few:
variants of Helly's, Tverberg's, colorful and fractional Helly theorems,
existence of weak $\varepsilon$-nets, $(p,q)$-theorems, \dots
Nevertheless, the original bounds on Radon's number are too large to be
widely applicable.
We improve the situation in the plane and show how to obtain polynomial bounds.
More generally the result extends to other two-dimensional (pseudo)manifolds.
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Extent |
27.0 minutes
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Subject | |
Type | |
File Format |
video/mp4
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Language |
eng
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Notes |
Author affiliation: IST Austria
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Series | |
Date Available |
2020-04-07
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0389761
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URI | |
Affiliation | |
Peer Review Status |
Unreviewed
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Scholarly Level |
Postdoctoral
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Rights URI | |
Aggregated Source Repository |
DSpace
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Item Media
Item Citations and Data
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International