×

zbMATH — the first resource for mathematics

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.

MSC:
52A40 Inequalities and extremum problems involving convexity in convex geometry
49Q20 Variational problems in a geometric measure-theoretic setting
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Math. Monogr., Oxford Univ. Press, New York (2000) · Zbl 0957.49001
[2] Almgren, F. J., Lieb, E. H.: Symmetric decreasing rearrangement is sometimes continuous. J. Amer. Math. Soc. 2, 683-773 (1989) · Zbl 0688.46014 · doi:10.2307/1990893
[3] Barvinok, A.: A Course in Convexity. Grad. Stud. Math. 54, Amer. Math. Soc., Providence (2002) · Zbl 1014.52001
[4] Brock, F., Solynin, A. Yu: An approach to symmetrization via polarization. Trans. Amer. Math. Soc. 352, 1759-1796 (2000) · Zbl 0965.49001 · doi:10.1090/S0002-9947-99-02558-1
[5] Chlebík, M., Cianchi, A., Fusco, N.: The perimeter inequality under Steiner symmetrization: cases of equality. Ann. of Math. 162, 525-555 (2005) · Zbl 1087.28003 · doi:10.4007/annals.2005.162.525 · euclid:annm/1134073590
[6] Chua, S.-K., Wheeden, R. L.: Weighted Poincaré inequalities on convex domains. Math. Res. Lett. 17, 993-1011 (2010) · Zbl 1226.46027 · doi:10.4310/MRL.2010.v17.n5.a15
[7] Cianchi, A., Fusco, N.: Functions of bounded variation and rearrangements. Arch. Ration. Mech. Anal. 165, 1-40 (2002) · Zbl 1028.49035 · doi:10.1007/s00205-002-0214-9
[8] Cicalese, M., Leonardi, G.: A selection principle for the sharp quantitative isoperimetric in- equality. Arch. Ration. Mech. Anal. 206, 617-643 (2012) · Zbl 1257.49045 · doi:10.1007/s00205-012-0544-1 · arxiv:1007.3899
[9] Drelichman, I., Durán, R. G.: Improved Poincaré inequalities with weights. J. Math. Anal. Appl. 347, 286-293 (2008) · Zbl 1154.46016 · doi:10.1016/j.jmaa.2008.06.005 · arxiv:0711.3399
[10] Esposito, L., Fusco, N., Trombetti, C.: A quantitative version of the isoperimetric inequal- ity: the anisotropic case. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 4, 619-651 (2005) · Zbl 1170.52300 · eudml:84574
[11] Figalli, A., Maggi, F., Pratelli, A.: A mass transportation approach to quantitative isoperimet- ric inequalities. Invent. Math. 182, 167-211 (2010) · Zbl 1196.49033 · doi:10.1007/s00222-010-0261-z
[12] Fuglede, B.: Stability in the isoperimetric problem for convex or nearly spherical domains in n R . Trans. Amer. Math. Soc. 314, 619-638 (1989) · Zbl 0679.52007 · doi:10.2307/2001401
[13] Fusco, N., Maggi, F., Pratelli, A.: The sharp quantitative isoperimetric inequality. Ann. of Math. 168, 941-980 (2008) · Zbl 1187.52009 · doi:10.4007/annals.2008.168.941 · annals.math.princeton.edu
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.