An Easy Reduction of an Isoperimetric Inequality on the Sphere to Extremal Set Theory

Abstract

In ancient times the extent of a city or an armed camp was often given in terms of its perimeter (so that a town would be described as requiring so many thousand paces to walk round). In the same way, according to Proclus, certain socialistic communities used to divide land so that each family received a plot of equal perimeter and it may have been in this context that it was first realised that a square contains a much greater area than a long thin rectangle of the same perimeter. Once it was understood that figures with the same perimeter may contain different areas, it was natural to ask whether there exists a figure of maximum area. It is not hard to guess that the answer is a circle, but a guess is not a proof. The isoperimetric problem thus asks for a proof that among all figures of equal perimeter the circle has greatest area.-T. W. Korner [3, ?35] The classical isoperimetric problem described by Korner has been dazzlingly generalized in numerous directions and the term is now used to describe a variety of conceptually similar problems. One particularly satisfying aspect of the study of such inequalities is the synergistic relationship between combinatorial and continuous cases of the problem. Indeed, in this note, we show how a combinatorial isoperimetric inequality on the n-hypercube HI, = {O, i} naturally gives rise to a continuous version on the n - 1 sphere. The specific question we explore is that of the largest volume that a (measur