A Variant of the Discrete Isoperimetric Problem

Abstract

We consider a variant of what is known as the discrete isoperimetric problem, namely the problem of minimising the size of the boundary of a family of subsets of a finite set. We use the technique of `shifting’ to provide an alternative proof of a result of Hart. This technique was introduced in the early $1980s$ by Frankl and Füredi and gave alternative proofs of previously known classical results like the discrete isoperimetric problem itself and the Kruskal-Katona theorem. Hence our purpose is to bring Hart’s result into this general framework.