# Isoperimetric problem

Among all closed plane curves with a given perimeter find one that maximizes the area. This is also known as Queen Dido's problem and serves as a classical problem for variational calculus. Its significance in mathematical programming is that it led to Lagrange's multiplier theorem. Restricting shapes to rectangles, the problem is:

$LaTeX: \max \{xy : x+y=p, \, x \ge 0, \, y \ge 0\},$

where $LaTeX: 2p$ is the perimeter. For positive $LaTeX: x$ and $LaTeX: y$ and multiplier $LaTeX: u$, Lagrange's multiplier conditions requires $LaTeX: y-u=0$ and $LaTeX: x-u=0$, so $LaTeX: x=y$, which means the solution is the square.