Variations of Helly's theorem for convex splinters

A convex splinter $K$ is a union of convex sets in ${\mathbb{R}}^d$ such that every minimal affine dependent set of ${\mathbb{R}}^d$ contained in $K$ is contained in one of the sets. The study of convex splinters was motivated by the study of flat transversals to convex sets. We extend several variations of Helly’s theorem from convex geometry to convex splinters. These include fractional and colorful variations of Helly’s theorem. We also extend Tverberg’s theorem to convex splinters.