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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.14663 |
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| _version_ | 1866914395914764288 |
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| author | Samarakkody, Miraj |
| author_facet | Samarakkody, Miraj |
| contents | 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 |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_14663 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Formalizing the Classical Isoperimetric Inequality in the Two-Dimensional Case Samarakkody, Miraj Metric Geometry Logic in Computer Science 53A, 53-04, 53-08 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 |
| title | Formalizing the Classical Isoperimetric Inequality in the Two-Dimensional Case |
| topic | Metric Geometry Logic in Computer Science 53A, 53-04, 53-08 |
| url | https://arxiv.org/abs/2603.14663 |