# Theorems of Carath'eodory, Helly, and Tverberg without dimension

arXiv: https://arxiv.org/abs/1806.08725 · HTML: https://mlsanity.com/paper/1806.08725/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Jun 2018.

## Built on by (most cited first, 4 of 4)

- [Towards the mathematical foundation of the minimum enclosing ball and related problems](https://mlsanity.com/paper/2402.06629/) (2024)
- [Hardness of High-Dimensional Linear Classification](https://mlsanity.com/paper/2603.19061/) (2026)
- [Towards the methodology for solving the minimum enclosing ball and related problems](https://mlsanity.com/paper/2410.11063/) (2024)
- [The VC dimension of partial concept classes via Radon's theorem](https://mlsanity.com/paper/2607.10751/) (2026)

## Abstract (arXiv)

We prove a no-dimensional version of Carath'edory's theorem: given an n-element set Psubset Red, a point a ∈ conv P, and an integer r≤ d, r ≤ n, there is a subset Qsubset P of r elements such that the distance between a and conv Q is less than diam P/√ 2r. A general no-dimension Helly type result is also proved with colourful and fractional consequences. Similar versions of Tverberg's theorem and some of their extensions are also established.