Theorems of Carathéodory, Helly, and Tverberg Without Dimension

Karim Adiprasito*, Imre Bárány, Nabil H. Mustafa, Tamás Terpai

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

We initiate the study of no-dimensional versions of classical theorems in convexity. One example is Carathéodory’s theorem without dimension: given an n-element set P in a Euclidean space, a point a∈convP, and an integer r≤ n, there is a subset Q⊂ P of r elements such that the distance between a and convQ is less than diamP/2r. In an analoguos no-dimension Helly theorem a finite family F of convex bodies is given, all of them are contained in the Euclidean unit ball of Rd. If k≤ d, | F| ≥ k, and every k-element subfamily of F is intersecting, then there is a point q∈ Rd which is closer than 1/k to every set in F. This result has several colourful and fractional consequences. Similar versions of Tverberg’s theorem and some of their extensions are also established.

Original languageAmerican English
Pages (from-to)233-258
Number of pages26
JournalDiscrete and Computational Geometry
Volume64
Issue number2
DOIs
StatePublished - 1 Sep 2020

Bibliographical note

Publisher Copyright:
© 2020, Springer Science+Business Media, LLC, part of Springer Nature.

Keywords

  • Carathéodory theorem
  • Convex approximation
  • Tverberg theorem

Fingerprint

Dive into the research topics of 'Theorems of Carathéodory, Helly, and Tverberg Without Dimension'. Together they form a unique fingerprint.

Cite this