Adiprasito, K;
Bárány, I;
Mustafa, NH;
Terpai, T;
(2020)
Theorems of Carathéodory, Helly, and Tverberg Without Dimension.
Discrete and Computational Geometry
, 64
pp. 233-258.
10.1007/s00454-020-00172-5.
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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 Carathéodory, Helly, and Tverberg Without Dimension |
| Open access status: | An open access version is available from UCL Discovery |
| 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. |
| Keywords: | Convex approximation, Tverberg theorem, Carathéodory theorem |
| UCL classification: | UCL 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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