Test

1. The Problem

Theorem 1 (Isoperimetric inequality in the plane).

Let $E$ be a compact domain in $\mathbb{R}^2$ with smooth boundary. Then \[|\partial E| \geq 2 \pi^{\frac{1}{2}} |E|^{\frac{1}{2}}.\] \end{theorem}

Here, $|E|$ denotes the area of $E$ and $|\partial E|$ denotes the length of the boundary $\partial E$. Note that disks achieve equality in the isoperimetric inequality. Indeed, if $E$ is a closed disk of radius $r$ in the plane, then $|E| = \pi r^2$ and $|\partial E| = 2\pi r$.

Theorem \ref{isoperimetric.inequality.2D} is a special case of a more general inequality which holds in arbitrary dimension.