---
_id: '1688'
abstract:
- lang: eng
text: 'We estimate the selection constant in the following geometric selection theorem
by Pach: For every positive integer d, there is a constant (Formula presented.)
such that whenever (Formula presented.) are n-element subsets of (Formula presented.),
we can find a point (Formula presented.) and subsets (Formula presented.) for
every i∈[d+1], each of size at least cdn, such that p belongs to all rainbowd-simplices
determined by (Formula presented.) simplices with one vertex in each Yi. We show
a super-exponentially decreasing upper bound (Formula presented.). The ideas used
in the proof of the upper bound also help us to prove Pach’s theorem with (Formula
presented.), which is a lower bound doubly exponentially decreasing in d (up to
some polynomial in the exponent). For comparison, Pach’s original approach yields
a triply exponentially decreasing lower bound. On the other hand, Fox, Pach, and
Suk recently obtained a hypergraph density result implying a proof of Pach’s theorem
with (Formula presented.). In our construction for the upper bound, we use the
fact that the minimum solid angle of every d-simplex is super-exponentially small.
This fact was previously unknown and might be of independent interest. For the
lower bound, we improve the ‘separation’ part of the argument by showing that
in one of the key steps only d+1 separations are necessary, compared to 2d separations
in the original proof. We also provide a measure version of Pach’s theorem.'
acknowledgement: R. K. was supported by the Russian Foundation for Basic Research
Grant 15-31-20403 (mol_a_ved) and grant 15-01-99563. J. K., Z. P., and M. T. were
partially supported by ERC Advanced Research Grant No. 267165 (DISCONV) and by the
project CE-ITI (GAČR P202/12/G061) of the Czech Science Foundation. J. K. was also
partially supported by Swiss National Science Foundation Grants 200021-137574 and
200020-14453. P. P., Z. P., and M. T. were partially supported by the Charles University
Grant GAUK 421511. P. P. was also partially supported by the Charles University
Grant SVV-2014-260107. Z. P. was also partially supported by the Charles University
Grant SVV-2014-260103.
author:
- first_name: Roman
full_name: Karasev, Roman
last_name: Karasev
- first_name: Jan
full_name: Kynčl, Jan
last_name: Kynčl
- first_name: Pavel
full_name: Paták, Pavel
last_name: Paták
- first_name: Zuzana
full_name: Patakova, Zuzana
last_name: Patakova
orcid: 0000-0002-3975-1683
- first_name: Martin
full_name: Tancer, Martin
id: 38AC689C-F248-11E8-B48F-1D18A9856A87
last_name: Tancer
orcid: 0000-0002-1191-6714
citation:
ama: Karasev R, Kynčl J, Paták P, Patakova Z, Tancer M. Bounds for Pach’s selection
theorem and for the minimum solid angle in a simplex. *Discrete & Computational
Geometry*. 2015;54(3):610-636. doi:10.1007/s00454-015-9720-z
apa: Karasev, R., Kynčl, J., Paták, P., Patakova, Z., & Tancer, M. (2015). Bounds
for Pach’s selection theorem and for the minimum solid angle in a simplex. *Discrete
& Computational Geometry*. Springer. https://doi.org/10.1007/s00454-015-9720-z
chicago: Karasev, Roman, Jan Kynčl, Pavel Paták, Zuzana Patakova, and Martin Tancer.
“Bounds for Pach’s Selection Theorem and for the Minimum Solid Angle in a Simplex.”
*Discrete & Computational Geometry*. Springer, 2015. https://doi.org/10.1007/s00454-015-9720-z.
ieee: R. Karasev, J. Kynčl, P. Paták, Z. Patakova, and M. Tancer, “Bounds for Pach’s
selection theorem and for the minimum solid angle in a simplex,” *Discrete &
Computational Geometry*, vol. 54, no. 3. Springer, pp. 610–636, 2015.
ista: Karasev R, Kynčl J, Paták P, Patakova Z, Tancer M. 2015. Bounds for Pach’s
selection theorem and for the minimum solid angle in a simplex. Discrete &
Computational Geometry. 54(3), 610–636.
mla: Karasev, Roman, et al. “Bounds for Pach’s Selection Theorem and for the Minimum
Solid Angle in a Simplex.” *Discrete & Computational Geometry*, vol.
54, no. 3, Springer, 2015, pp. 610–36, doi:10.1007/s00454-015-9720-z.
short: R. Karasev, J. Kynčl, P. Paták, Z. Patakova, M. Tancer, Discrete & Computational
Geometry 54 (2015) 610–636.
date_created: 2018-12-11T11:53:28Z
date_published: 2015-10-01T00:00:00Z
date_updated: 2021-01-12T06:52:32Z
day: '01'
department:
- _id: UlWa
doi: 10.1007/s00454-015-9720-z
intvolume: ' 54'
issue: '3'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1403.8147
month: '10'
oa: 1
oa_version: Preprint
page: 610 - 636
publication: Discrete & Computational Geometry
publication_status: published
publisher: Springer
publist_id: '5457'
quality_controlled: '1'
scopus_import: 1
status: public
title: Bounds for Pach's selection theorem and for the minimum solid angle in a simplex
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 54
year: '2015'
...