## 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 R^{d}. If k≤ d, | F| ≥ k, and every k-element subfamily of F is intersecting, then there is a point q∈ R^{d} 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 language | American English |
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Pages (from-to) | 233-258 |

Number of pages | 26 |

Journal | Discrete and Computational Geometry |

Volume | 64 |

Issue number | 2 |

DOIs | |

State | Published - 1 Sep 2020 |

### Bibliographical note

Funding Information:K.A. was supported by ERC StG 716424-CASe and ISF Grant 1050/16. I.B. was supported by the Hungarian National Research, Development and Innovation Office NKFIH Grants K 111827 and K 116769, and by ERC-AdG 321104. N.M. was supported by the grant ANR SAGA (JCJC-14-CE25-0016-01). T.T. was supported by the Hungarian National Research, Development and Innovation Office NKFIH Grants NK 112735 and K 120697.

Publisher Copyright:

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

## Keywords

- Carathéodory theorem
- Convex approximation
- Tverberg theorem