Formalizing the Classical Isoperimetric Inequality in the Two-Dimensional Case

Abstract

We present a formal verification of the classical isoperimetric inequality in the plane using the Lean 4 proof assistant and its mathematical library Mathlib. We follow Adolf Hurwitz's analytic approach to establish the inequality $L^2 \ge 4πA$, which states that among all simple closed curves of a given perimeter $L$, the circle uniquely maximizes the enclosed area $A$. The formalization proceeds in two phases. In the first phase, we establish the Fourier-analytic foundations required by Hurwitz's approach: we formalize orthogonality relations for trigonometric functions over $[-π,π]$, Parseval's theorem for classical Fourier series, uniform convergence of Fourier partial sums via the Weierstrass M-test, term-by-term differentiability, and Wirtinger's inequality. In the second phase, we carry out Hurwitz's proof itself: working with simple closed $C^1$ curves given in arc-length parametrization, we reparametrize over $[0,2π]$, establish the shoelace area formula, apply integration by parts, invoke the AM--GM inequality, apply Wirtinger's inequality, and use the arc-length constraint to derive the bound $A \le L^2/(4π)$. We discuss the key formalization challenges encountered, including the interchange of infinite sums and integrals, term-by-term differentiation, and the coordination of different indexing conventions within Mathlib. The complete formalization is available at https://github.com/mirajcs/IsoperimetricInequality