A variant of the discrete isoperimetric problem

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 $\lq$shifting' to provide an alternative proof of a result of Hart. This technique was introduced in the early 1980s by Frankl and F\"{u}redi 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