A Mélange of Diameter Helly-Type Theorems

@article{Dillon2021AMO,
  title={A M{\'e}lange of Diameter Helly-Type Theorems},
  author={Travis Dillon and Pablo Sober'on},
  journal={SIAM J. Discret. Math.},
  year={2021},
  volume={35},
  pages={1615-1627}
}
A Helly-type theorem for diameter provides a bound on the diameter of the intersection of a finite family of convex sets in $\mathbb{R}^d$ given some information on the diameter of the intersection of all sufficiently small subfamilies. We prove fractional and colorful versions of a longstanding conjecture by Barany, Katchalski, and Pach. We also show that a Minkowski norm admits an exact Helly-type theorem for diameter if and only if its unit ball is a polytope and prove a colorful version for… 
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