Showing the Surprising Difficulty of Proving That a Circle has the Smallest Perimeter for a Given Area, and Other Interesting Related Problems

Abstract

Throughout this paper, I shall show why a circle has the minimum perimeter for a given area, using the isoperimetric inequality, and see if this is possible for a sphere. We shall also briefly discuss an unsolved problem which relates quite well to this topic and if solved could show, in a nice way, that a sphere has the smallest ratio between surface area and volume.