## Stability in the isoperimetric problem for convex or nearly spherical domains in $${\mathbb{R}}^ n$$.(English)Zbl 0679.52007

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.
Reviewer: V.Anisiu

### MSC:

 52A40 Inequalities and extremum problems involving convexity in convex geometry 52A20 Convex sets in $$n$$ dimensions (including convex hypersurfaces) 52A15 Convex sets in $$3$$ dimensions (including convex surfaces) 49Q20 Variational problems in a geometric measure-theoretic setting
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