For a compact domain D in \({\mathbb{R}}^ n\) with Lipschitz class boundary \(\partial D\), V and S denote the volume of D and the area of D respectively. If v and s are defined by: \(V=\omega_ nv^ n\), \(S=n\omega_ ns^{n-1}\) where \(\omega_ n\) denotes the volume of the unit ball in \({\mathbb{R}}^ n\), \(\Delta =(s/v)^{n-1}-1\) is the isoperimetric deficiency of D.
After a change of scale, D can be represented in polar coordinates: \(R(\xi)=1+u(\xi)\), \(\xi\in \Sigma\), where \(\Sigma\) is the unit sphere in \({\mathbb{R}}^ n.\)
The isoperimetric inequality \(\Delta\geq 0\) is refined by: \[ (1/10)(\| u\|^ 2_ 2+\| \nabla u\|^ 2_ 2)\leq \Delta \leq (3/5)\| \nabla u|^ 2_ 2 \] valid for near spherical domains (i.e. for which \(\| u\|_{\infty}\leq 3/2on\), \(\| \nabla u\|_{\infty}\leq 1/2).\)
For convex bodies, estimations of the form \(\| u\|_{\infty}\leq f(\Delta)\) are obtained, where f is an explicit elementary function vanishing continuously in 0.