  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 B\'ar\'any, 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 those that do. Finally, we
prove Helly-type theorems for the property of ``containing $k$ colinear integer
points.
