Page 166 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Chapter 6

Isoperimetric inequality

6.1 Introduction

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Let A ⊂ R be a bounded open set whose boundary, ∂A,is a sufficiently regular,
simple closed curve. Denote by L (∂A) the length of the boundary and by M (A)
the measure (the area) of A. The isoperimetric inequality states that

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[L (∂A)] − 4πM (A) ≥ 0 .

Furthermore, equality holds if and only if A is a disk (i.e., ∂A is a circle).
This is one of the oldest problems in mathematics. A variant of this in-
equality is known as Dido problem (who is said to have been a Phoenician
princess). Several more or less rigorous proofs were known since the times of the
Ancient Greeks; the most notable attempt for proving the inequality is due to
Zenodorus, who proved the inequality for polygons. There are also significant
contributions by Archimedes and Pappus. To come closer to us one can mention,
among many, Euler, Galileo, Legendre, L’Huilier, Riccati or Simpson. A special
tribute should be paid to Steiner who derived necessary conditions through a
clever argument of symmetrization. The first proof that agrees with modern
standards is due to Weierstrass. Since then, many proofs were given, notably by
Blaschke, Bonnesen, Carathéodory, Edler, Frobenius, Hurwitz, Lebesgue, Lieb-
mann, Minkowski, H.A. Schwarz, Sturm, and Tonelli among others. We refer to
Porter [86] for an interesting article on the history of the inequality.
We will give here the proof of Hurwitz as modified by H. Lewy and Hardy-
Littlewood-Polya [55]. In particular we will show that the isoperimetric inequal-
ity is equivalent to Wirtinger inequality that we have already encountered in a

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