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A quantitative isoperimetric inequality in \(n\)-dimensional space. (English) Zbl 0746.52012
This paper is about the question of stability in the isoperimetric inequality. The surface area of a (measurable) set \(E\subseteq \mathbb{R}^ n\) exceeds that of a ball of similar volume by a factor \(1+\varepsilon\), (\(\varepsilon \geq 0\)) and it is shown that if \(\varepsilon\) is small the \(E\) is almost a ball in the sense that its Fraenkel asymmetry \(\lambda(E)\) is small: precisely \(\lambda(E)^ 4=O(\varepsilon)\) where the constant, given explicitly, depends on \(n\). The conjectured sharp result would have exponent 2 instead of 4. No constraint such as convexity or connectedness is imposed on E. Recent papers on this subject are R. Osserman, Complex Variables 9, 241-249 (1987; Zbl 0602.52006)] and B. Fuglede, Trans. Am. Math. Soc. 314, No. 2, 619- 638 (1989; Zbl 0679.52007)]. An application in potential theory is possible via the method of R. R. Hall, W. Hayman and A. W. Weitsman, J. Anal. Math. 56, 87-123 (1991)], to bound the asymmetry in terms of the excess capacity of \(E\) (compared to the ball).
Reviewer: R.R.Hall

52A40 Inequalities and extremum problems involving convexity in convex geometry
49Q20 Variational problems in a geometric measure-theoretic setting
52A38 Length, area, volume and convex sets (aspects of convex geometry)
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
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