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Piercing intersecting convex sets

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications Pub Date : 2025-02-05 DOI:10.1016/j.laa.2025.02.007

Imre Bárány , Travis Dillon , Dömötör Pálvölgyi , Dániel Varga

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引用次数: 0

Abstract

Assume two finite families

A

and

B

of convex sets in

R^{3}

have the property that

A \cap B \neq \emptyset

for every

A \in A

and

B \in B

. Is there a constant

γ > 0

(independent of

A

and

B

) such that there is a line intersecting

γ | A |

sets in

A

γ | B |

sets in

B

? This is an intriguing Helly-type question from a paper by Martínez, 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.

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来源期刊

Linear Algebra and its Applications 数学-应用数学

CiteScore

2.20

自引率

9.10%

发文量

333

审稿时长

13.8 months

期刊介绍： Linear Algebra and its Applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra in their algebraic, arithmetic, combinatorial, geometric, or numerical aspects. It also publishes articles that give significant applications of matrix theory or linear algebra to other branches of mathematics and to other sciences. Articles that provide new information or perspectives on the historical development of matrix theory and linear algebra are also welcome. Expository articles which can serve as an introduction to a subject for workers in related areas and which bring one to the frontiers of research are encouraged. Reviews of books are published occasionally as are conference reports that provide an historical record of major meetings on matrix theory and linear algebra.