### Isoperimetric Sets of Integers

By Edward Newkirk

November 2, 2011

#### Abstract

It's a well-known fact that the circle has the least perimeter of all planar regions enclosing given area. Similar minimizers have been studied on many other Riemannian manifolds; this area of inquiry is referred to as the isoperimetric problem. We will discuss the isoperimetric problem on the integers, find an asymptotic limit for the perimeter required to enclose given area, and also incidentally explain how on earth we can talk about "area" and "perimeter" for integers.