On the cardinality of sets in ${\bf R}^d$ obeying a slightly obtuse angle bound

Tongseok Lim; Robert J. McCann

On the cardinality of sets in ${\bf R}^d$ obeying a slightly obtuse angle bound

Metric Geometry 2022-02-03 v4 Combinatorics Optimization and Control

Authors: Tongseok Lim , Robert J. McCann

Abstract

In this paper we explicitly estimate the number of points in a subset $A \subset \R^{d}$ as a function of the maximum angle $\angle A$ that any three of these points form, provided $\angle A < \theta_d := \arccos(-\frac 1 {d}) \in (\pi/2,\pi)$ . We also show $\angle A < \theta_d$ ensures that $A$ coincides with the vertex set of a convex polytope. This study is motivated by a question of Paul Erd\H{o}s and indirectly by a conjecture of L\'aszl\'o Fejes T\'oth.

Keywords

point sets obtuse angle bound cardinality convex polytope euclidean space

Cite

@article{arxiv.2007.13871,
  title  = {On the cardinality of sets in ${\bf R}^d$ obeying a slightly obtuse angle bound},
  author = {Tongseok Lim and Robert J. McCann},
  journal= {arXiv preprint arXiv:2007.13871},
  year   = {2022}
}

Comments

v4 will be published in the SIAM Journal on Discrete Mathematics

On the cardinality of sets in ${\bf R}^d$ obeying a slightly obtuse angle bound

Abstract

Keywords

Cite

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