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Stability of the Steiner symmetrization of convex sets. (English) Zbl 1277.52012
The paper is concerned with the Steiner symmetrization of any codimension. The general case \(1\leq k\leq n-1\) is examined using a new approach based on the regularity properties of the barycenter of the sections \(E_{x'}:=\{y\in \mathbb R^k\mid (x',y)\in E\}\) as \(x'\) varies in \(\pi^+(E):=\{x'\in \mathbb R^{n-k}\mid L(x')>0\}\), \(L(x'):=\mathcal L^k(E_{x'})\), \(\mathcal L^k\) stands for the outer Lebesgue measure in \(\mathbb R^k\). The advantage of this approach is twofold. Firstly, it proves possible to recover and extend the result of 2005 by M. Chlebík et al. [Ann. Math. (2) 162, No. 1, 525–555 (2005; Zbl 1087.28003)] for \(k = 1\) to any codimension, with a new and simpler proof. Secondly, one now is able to obtain a quantitative isoperimetric estimate for convex sets which, to the best of authors’ knowledge, is the first result of this kind in the framework of Steiner symmetrization.

52A40 Inequalities and extremum problems involving convexity in convex geometry
49Q20 Variational problems in a geometric measure-theoretic setting
Full Text: DOI
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