Oscar Leong, Mathematics

Helly's Theorem and its Generalization to the Union of Convex Sets

In 1913, Eduard Helly proved Helly’s Theorem, a statement about the intersection of a finite family of convex sets in R^d given a condition on the minimum number of sets in a finite subfamily whose intersection is non-empty. This theorem spawned a number of generalizations and variants that may be categorized as Helly-Type theorems. Simply put, a Helly-Type theorem states that if every n members of a family of objects have a particular property T, then the entire family of objects has the property T. This presentation is a survey on Helly-Type theorems that involve the finite union of convex sets and will discuss the implications and significance of these types of mathematical generalizations to mathematics as a whole.