isoperimetric problem
/ eye-so-peh-rih-MET-rik /
You have a fixed length of fence. What is the largest area of field you can enclose with it? The classical answer, known since antiquity and tied to the legend of Queen Dido cutting an oxhide into thin strips, is the circle: among all closed curves of a given perimeter, the circle encloses the most area. The isoperimetric (equal-perimeter) problem is the prototype of constrained variational problems — optimize one functional (area) while another functional (perimeter, or length) is held fixed.
The structure is what matters: you maximize an area functional subject to a fixed-length constraint. The tool is a Lagrange multiplier for functionals: you form a combined integrand (area integrand) minus lambda times (length integrand), apply the Euler-Lagrange equation to that combination, and the single unknown constant lambda is later pinned down by the length constraint. Carrying this out for the plane gives a curve of constant curvature — a circle of radius set by lambda. The companion isoperimetric inequality states it as a clean bound: for any closed plane curve, 4 pi times area is at most the perimeter squared, with equality only for the circle.
The name 'isoperimetric' has since become the generic label for any variational problem with an integral constraint, even when no perimeter is involved — the hanging-chain (catenary) problem, where energy is minimized at fixed length, is an isoperimetric problem in this broad sense. The circle result generalizes to higher dimensions (the sphere encloses the most volume for given surface area, which is why bubbles are round and why a planet pulled together by gravity is nearly spherical) and underlies much of geometric analysis. An honest note: an isoperimetric extremal solves the multiplier equation, but you must still confirm it is the maximizer (not a minimizer or saddle) — the circle is genuinely the maximizer here, but the Euler-Lagrange step alone does not prove it.
With 100 meters of fence, a circle encloses about 796 square meters (radius 100/(2 pi) ≈ 15.9 m), while a square of the same perimeter encloses only 625 square meters. Same fence, more land — the circle wins, exactly as the isoperimetric inequality predicts.
For a fixed perimeter the circle encloses the maximum area; the square falls short.
'Isoperimetric problem' is now a general name for any variational problem with an integral constraint, not only perimeter-versus-area. The defining feature is the fixed integral side-condition handled by a multiplier.