The isoperimetric problem in euclidean space

Abstract

Fleeing the vengeance of her brother, Dido lands on the coast of North Africa and founds the city of Carthage. Within the mythology associated with Virgil’s saga lies one of the earliest problems in extremal geometric analysis. For the bargain which Dido agrees to with a local potentate is this: she may have that portion of land which she is able to enclose with the hide of a bull. Legend records Dido’s ingenious and elegant solution: cutting the hide into a series of long thin strips, she marks out a vast circumference, forming the eventual line of the walls of ancient Carthage. This problem is a variant of what has become known as the classical isoperimetric problem.1 In more precise terms it may be formulated as follows: among all bounded, connected open regions in the plane with a fixed perimeter, characterize those regions with the maximal volume. Needless to say, Dido’s solution is correct: the extremal regions are precisely open circular planar discs.