Isoperimetric problem

Isoperimetric problem
Oleg Alexandrov · Public domain · source
Name	Isoperimetric problem
Field	Mathematics
Solved	Ancient

Isoperimetric problem is a classical optimization question about finding shapes of largest area enclosed by a closed curve of given perimeter. Originating in antiquity and recurring through Euclid, Archimedes, Pythagoras of Samos, Hippocrates of Chios, this problem has influenced developments in Calculus of variations, Geometric measure theory, Differential geometry, Convex geometry, and Mathematical physics. Connections extend to figures studied by Leonhard Euler, Joseph-Louis Lagrange, Lord Kelvin, Henri Poincaré, and modern work by Ennio De Giorgi, John Nash, William Thurston, and Terence Tao.

History

Ancient accounts credit inquiries during the era of Plato, Aristotle, Euclid, and Archimedes about optimal enclosures, with medieval treatments in the period of Alhazen and early modern examinations by Kepler, Descartes, Pascal, and Fermat. The eighteenth century saw formal analysis by Euler, Lagrange, and Jacques-Louis Lagrange leading into the nineteenth-century rigorization by Cauchy, Weierstrass, Riemann, Dirichlet, and Gauss. The proof for the plane circle as optimal was structured using methods later associated with Isabelle, Hilbert, Noether, and culminated in modern frameworks developed by Sofia Kovalevskaya and David Hilbert in the context of variational principles and Plateau's problem analogies.

Mathematical formulation

Formulations appear in Euclidean, spherical, and hyperbolic settings and are stated as constrained extrema: maximize area given fixed boundary length or, conversely, minimize perimeter given fixed enclosed area. In analytical terms one frames the problem on function spaces studied by Bernhard Riemann, Georg Cantor, Emmy Noether, and Andrey Kolmogorov and casts constraints using Lagrange multipliers introduced by Joseph-Louis Lagrange and generalized via the Euler–Lagrange equation central to Leonhard Euler's work. Modern variational formulations use measure-theoretic tools by Ennio De Giorgi, Lars Hörmander, and John von Neumann and exploit compactness theorems connected to results by Stefan Banach, Felix Hausdorff, and Henri Lebesgue.

Solutions and methods

Classical solutions identify the circle (in the plane), spherical caps (on the sphere), and geodesic disks (in hyperbolic space) as extremals; proofs draw on symmetrization techniques developed by Józef Pólya, George Pólya, Gábor Szegő, and Stefan Banach and on integral inequalities associated with Augustin-Louis Cauchy, Inequalities by Markov, and Szegő. The direct method in the calculus of variations attributed to David Hilbert and formalized by Ennio De Giorgi yields existence results, while regularity and uniqueness use techniques from Federer, Herbert Federer, Almgren, and Jean Taylor. Geometric flows like the curve shortening flow and methods related to the Ricci flow studied by Richard Hamilton and Grigori Perelman provide alternative approaches; variational inequalities and rearrangement methods tie to work by Marcel Riesz, Hardy Littlewood, and Emil Artin.

Variants and generalizations

Higher-dimensional analogues produce the isoperimetric inequality in n-dimensional Euclidean space studied by Henri Lebesgue, Joseph Fourier, Georges Birkhoff, Michael Atiyah, and Isadore Singer; anisotropic and weighted versions connect to research by Gian-Carlo Rota and Cecilia Rota and to functional inequalities explored by Elliott Lieb, Michael Loss, and Larry Guth. Discrete and combinatorial variants appear in graph theory and network design with contributions from Paul Erdős, Alfréd Rényi, Erdős–Rényi model, and László Lovász; geometric measure generalizations intersect with Minimal surface problems studied by Jesse Douglas, Ennio De Giorgi, and Richard Schoen. Constrained settings include prescribed curvature studied by Aleksandr Aleksandrov, Andrei Kolmogorov, and capillarity problems with links to Joseph Plateau and Thomas Young.

Applications and related problems

The isoperimetric paradigm informs optimal design in physics and engineering problems addressed by Lord Kelvin, James Clerk Maxwell, Marie Curie, and Ludwig Boltzmann; it underpins models in capillarity and surface tension studied by Joseph Plateau and Pierre-Simon Laplace and appears in biological morphology discussed by Charles Darwin and D'Arcy Thompson. In probability and analysis it relates to concentration of measure theorems by Paul Lévy, Vitaly Milman, Mikhail Gromov, and Gilles Pisier and impacts spectral geometry through the Weyl law and results of Hermann Weyl, Atle Selberg, and Mark Kac. Computational and discrete applications connect to algorithms in Donald Knuth's work, optimization studied by George Dantzig, and network flow problems by L. R. Ford Jr. and D. R. Fulkerson, while connections to modern topology and geometry reflect themes in the work of William Thurston, Michael Freedman, and Shing-Tung Yau.

Category:Geometry