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}
}
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. 
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Isometric and affine copies of a set in volumetric Helly results
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<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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References

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A new variation of Tverberg’s theorem is presented, given a discrete set S, and the number of points needed to guarantee the existence of an m-partition of the points such that the intersection of the m convex hulls of the parts contains at least k points of S.
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