# Quantitative Combinatorial Geometry for Continuous Parameters

@article{Loera2017QuantitativeCG,
title={Quantitative Combinatorial Geometry for Continuous Parameters},
author={Jes{\'u}s A. De Loera and Reuben N. La Haye and David Rolnick and Pablo Sober{\'o}n},
journal={Discrete \& Computational Geometry},
year={2017},
volume={57},
pages={318-334}
}
• Published 17 March 2016
• Mathematics
• Discrete & Computational Geometry
We prove variations of Carathéodory’s, Helly’s and Tverberg’s theorems where the sets involved are measured according to continuous functions such as the volume or diameter. Among our results, we present continuous quantitative versions of Lovász’s colorful Helly’s theorem, Bárány’s colorful Carathéodory’s theorem, and the colorful Tverberg’s theorem.
12 Citations
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Bulletin of the American Mathematical Society
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Tverberg’s theorem is 50 years old: A survey
• Mathematics
Bulletin of the American Mathematical Society
• 2018
This survey presents an overview of the advances around Tverberg’s theorem, focusing on the last two decades. We discuss the topological, linear-algebraic, and combinatorial aspects of Tverberg’s
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SIAM J. Discret. Math.
• 2021
It is shown that a Minkowski norm admits an exact Helly-type theorem for diameter if and only if its unit ball is a polytope and a colorful version for those that do is proved.
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Define the k-th Radon number rk of a convexity space as the smallest number (if it exists) for which any set of rk points can be partitioned into k parts whose convex hulls intersect. Combining the
Radon Numbers Grow Linearly
Combining the recent abstract fractional Helly theorem of Holmsen and Lee with earlier methods of Bukh, it is proved that r_k grows linearly, i.e., $r_k\le c(r_2)\cdot k$.
Radon Numbers Grow Linearly
<jats:p>Define the <jats:italic>k</jats:italic>-th Radon number <jats:inline-formula><jats:alternatives><jats:tex-math>$$r_k$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">

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