M.Sc Thesis

M.Sc StudentKhmelnitsky Igor
SubjectD-collapsibility and its applications
DepartmentDepartment of Mathematics
Supervisor PROF. Roy Meshulam


A classical result of Helly asserts that given a finite family {K1, K2,?, Kn} of convex sets in Rd such that an intersection of any sub family of size<d is not empty, then the intersection of the entire family is not empty. Helly’s theorem and its many extensions and generalizations form a central area of study in discrete geometry and its applications. A key ingredient that plays an important role in a variety of Helly type theorems is the notion of d-collapsibility.

In this work we will define d-collapsibility as a d-dimensional extension of the notion of chordality in graphs. Following the review of some basic definitions and examples, we will present a number of results we obtained concerning the combinatorics of d-collapsible complexes. These include estimates on the collapsibility of intersection, union and joins of complexes. We conclude this work by relating d-collapsibility to an old result by Vorobév concerning consistent measures and their extensions.