Helly‐type theorems for the diameter

We study versions of Helly's theorem that guarantee that the intersection of a family of convex sets inR dhas 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.