On the number of vertices of projective polytopes

IF 0.8 3区数学 Q2 MATHEMATICS

Mathematika Pub Date : 2023-03-23 DOI:10.1112/mtk.12193

Natalia García-Colín, Luis Pedro Montejano, Jorge Luis Ramírez Alfonsín

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引用次数: 1

Abstract

Let X be a set of n points in $R^{d}$ in general position. What is the maximum number of vertices that $conv (T (X))$ can have among all the possible permissible projective transformations T? In this paper, we investigate this and other related questions. After presenting several upper bounds, obtained by using oriented matroid machinery, we study a closely related problem (via Gale transforms) concerning the maximal number of minimal Radon partitions of a set of points. The latter led us to a result supporting a positive answer to a question of Pach and Szegedy asking whether balanced 2-colorings of points in the plane maximize the number of induced multicolored Radon partitions. We also discuss a related problem concerning the size of topes in arrangements of hyperplanes as well as a tolerance-type problem of finite sets.

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关于投影多面体的顶点数

设X是一般位置上Rd$\mathbb {R}^d$中n个点的集合。conv(T(X))$\mathsf {conv}(T(X))$在所有可能允许的投影变换T中可以拥有的顶点的最大数量是多少?在本文中，我们对这一问题和其他相关问题进行了研究。在给出了用定向矩阵机制得到的上界之后，我们研究了一个密切相关的问题(通过Gale变换)，即点集的最小Radon分区的最大数目。后者使我们得到了一个结果，支持Pach和Szegedy的一个问题的肯定答案，即平面上点的平衡2色是否最大化了诱导的多色Radon分区的数量。我们还讨论了一个有关超平面排列中类型大小的相关问题以及有限集的容差型问题。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Mathematika MATHEMATICS, APPLIED-MATHEMATICS

CiteScore

1.40

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Mathematika publishes both pure and applied mathematical articles and has done so continuously since its founding by Harold Davenport in the 1950s. The traditional emphasis has been towards the purer side of mathematics but applied mathematics and articles addressing both aspects are equally welcome. The journal is published by the London Mathematical Society, on behalf of its owner University College London, and will continue to publish research papers of the highest mathematical quality.