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A Helly-type theorem for diameter provides a bound on the diameter of theintersection of a finite family of convex sets in $\mathbb{R}^d$ given someinformation on the diameter of the intersection of all sufficiently smallsubfamilies. We prove fractional and colorful versions of a longstandingconjecture by B\'ar\'any, Katchalski, and Pach. We also show that a Minkowskinorm admits an exact Helly-type theorem for diameter if and only if its unitball is a polytope and prove a colorful version for those that do. Finally, weprove Helly-type theorems for the property of ``containing $k$ colinear integerpoints.

A m\'elange of diameter Helly-type theorems