The Isoperimetric Inequality on Natural Subsets

The isoperimetric problem is an exercise of classical geometry posing the following question. If a closed Jordan region on the plane has area A, what is the smallest perimeter that the gure can attain? This question was solved, yet recently an interesting reformulation of the question was posed. By viewing sets of natural numbers as objects, volume was defined as the sum of a sets elements, while perimeter was defined as the sum of all elements in a set with adjacent numbers not contained in a set. This new isoperimetric problem over the naturals then posed the question, If a subset of 0,1,2,... has volume n, what is the smallest possible value of its perimeter. In this thesis we seek to create tight bounds on this perimeter function, as well as construct an explicit set of minimal perimeter for all natural numbers.