Piercing intersecting convex sets

被引：0

作者：

Barany, Imre ^{[1
,2
]}

Dillon, Travis ^{[3
]}

Palvolgyi, Domotor ^{[1
,4
]}

Varga, Daniel ^{[1
]}

机构：

[1] HUN REN Alfred Reny Inst Math, 13 Realtanoda St, H-1053 Budapest, Hungary

[2] UCL, Dept Math, Gower St, London WC1E 6BT, England

[3] MIT, 77 Massachusetts Ave, Cambridge, MA USA

[4] Eotvos Lorand Univ, Budapest, Hungary

来源：

LINEAR ALGEBRA AND ITS APPLICATIONS | 2025年 / 710卷

基金：

芬兰科学院; 欧洲研究理事会; 美国国家科学基金会;

关键词：

Helly-type theorems; Line transversals; Linear programming;

D O I：

10.1016/j.laa.2025.02.007

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

Assume two finite families A and B of convex sets in R-3 have the property that A boolean AND B not equal & empty; for every A is an element of A and B is an element of B. Is there a constant gamma > 0 (independent of A and B) such that there is a line intersecting gamma|A| sets in A or gamma|B|sets in B? This is an intriguing Helly-type question from a paper by Martinez, Roldan and Rubin. We confirm this in the special case when all sets in A lie in parallel planes and all sets in B lie in parallel planes; in fact, one of the two families has a transversal by a single line. (c) 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).

引用

页码：405 / 417

页数：13

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