讲座内容：

This lecture focuses on Helly-type theorems and their applications in combinatorial geometry. It begins with a discussion of the Helly property for subtrees of a tree: if every pair of subtrees intersects, then all of them have a common point. The lecture then moves to finite families of convex sets in R^d, emphasizing the classical Helly idea that global intersection can be tested through small subfamilies. A central part of the lecture is a new proof of Helly’s Theorem using the Multiple Separation Theorem, with the important observation that it is enough to work with halfspaces. The final part of the lecture presents several applications, including Kirchberger’s Theorem, Rado’s Center Point Theorem, Jung’s theorem on enclosing balls, results on chords of convex bodies, and vertical segments in R^d.