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Theorems of Caratheodory, Helly, and Tverberg Without Dimension

Adiprasito, K; Barany, I; Mustafa, NH; Terpai, T; (2020) Theorems of Caratheodory, Helly, and Tverberg Without Dimension. Discrete and Computational Geometry 10.1007/s00454-020-00172-5. (In press).

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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.

Type: Article
Title: Theorems of Caratheodory, Helly, and Tverberg Without Dimension
DOI: 10.1007/s00454-020-00172-5
Publisher version: https://doi.org/10.1007/s00454-020-00172-5
Language: English
Additional information: This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions.
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: https://discovery.ucl.ac.uk/id/eprint/10105331
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