Helly-type theorems for the diameter
Pablo Soberón · Bulletin of the London Mathematical Society · 2016
We study versions of Helly's theorem that guarantee that the intersection of a family of convex sets in R d has a large diameter. This includes colourful, fractional and ( p , q ) versions of Helly's theorem. In particular, the fractional and ( p , q ) versions work with conditions where the corresponding Helly's theorem does not. We also include variants of Tverberg's theorem, Bárány's point selection theorem and the existence of weak epsilon-nets for convex sets with diameter estimates.