Transversals of families of sets in $\mathbb{R}^n$ and a connection between the Helly and Borsuk theorems

A criterion is obtained for the existence, given a family of convex sets in $\mathbb{R}^n$, of an $m$-dimensional plane intersecting all members of the family. The results are a generalization of the theorems of Helly, Horn–Klee, and Borsuk. Also presented are applications of these results to the geometry of convex sets and to combinatorics.