On Helly Numbers of Exponential Lattices

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F23%3A10473478" target="_blank" >RIV/00216208:11320/23:10473478 - isvavai.cz</a>

Result on the web

<a href="https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.8" target="_blank" >https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.8</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.4230/LIPIcs.SoCG.2023.8" target="_blank" >10.4230/LIPIcs.SoCG.2023.8</a>

Result language

angličtina

Original language name

On Helly Numbers of Exponential Lattices

Original language description

Given a set S SUBSET OF OR EQUAL TO   ℝ2, define the Helly number of S, denoted by H(S), as the smallest positive integer N, if it exists, for which the following statement is true: for any finite family ℱ of convex sets in ℝ2 such that the intersection of any N or fewer members of ℱ contains at least one point of S, there is a point of S common to all members of ℱ. We prove that the Helly numbers of exponential lattices {αn : n ELEMENT OF ℕo}2 are finite for every α &gt; 1 and we determine their exact values in some instances. In particular, we obtain H({2n : n ELEMENT OF ℕo}2) = 5, solving a problem posed by Dillon (2021). For real numbers α, β &gt; 1, we also fully characterize exponential lattices L(α,β) = {αn : n ELEMENT OF ℕo} x {βn : n ELEMENT OF ℕo} with finite Helly numbers by showing that H(L(α,β)) is finite if and only if log_α(β) is rational.

Czech name

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Czech description

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Type

D - Article in proceedings

CEP classification

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OECD FORD branch

10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)

Project

<a href="/en/project/GA21-32817S" target="_blank" >GA21-32817S: Algorithmic, structural and complexity aspects of geometric configurations</a><br>

Continuities

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year

2023

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Article name in the collection

39th International Symposium on Computational Geometry (SoCG 2023)

ISBN

978-3-95977-273-0

ISSN

1868-8969

e-ISSN

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Number of pages

16

Pages from-to

1-16

Publisher name

Neuveden

Place of publication

USA

Event location

Dallas

Event date

Jun 12, 2023

Type of event by nationality

WRD - Celosvětová akce

UT code for WoS article

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