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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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