Quantitative Isoperimetric Inequalities on the Real Line

Yohann de Castro[1]

  • [1] Institut de Mathématiques de Toulouse (UMR CNRS 5219) Université de Toulouse III - Paul Sabatier 118, route de Narbonne, 31062 Toulouse, France.

Annales mathématiques Blaise Pascal (2011)

  • Volume: 18, Issue: 2, page 251-271
  • ISSN: 1259-1734

Abstract

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In a recent paper A. Cianchi, N. Fusco, F. Maggi, and A. Pratelli have shown that, in the Gauss space, a set of given measure and almost minimal Gauss boundary measure is necessarily close to be a half-space.Using only geometric tools, we extend their result to all symmetric log-concave measures on the real line. We give sharp quantitative isoperimetric inequalities and prove that among sets of given measure and given asymmetry (distance to half line, i.e. distance to sets of minimal perimeter), the intervals or complements of intervals have minimal perimeter.

How to cite

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de Castro, Yohann. "Quantitative Isoperimetric Inequalities on the Real Line." Annales mathématiques Blaise Pascal 18.2 (2011): 251-271. <http://eudml.org/doc/219759>.

@article{deCastro2011,
abstract = {In a recent paper A. Cianchi, N. Fusco, F. Maggi, and A. Pratelli have shown that, in the Gauss space, a set of given measure and almost minimal Gauss boundary measure is necessarily close to be a half-space.Using only geometric tools, we extend their result to all symmetric log-concave measures on the real line. We give sharp quantitative isoperimetric inequalities and prove that among sets of given measure and given asymmetry (distance to half line, i.e. distance to sets of minimal perimeter), the intervals or complements of intervals have minimal perimeter.},
affiliation = {Institut de Mathématiques de Toulouse (UMR CNRS 5219) Université de Toulouse III - Paul Sabatier 118, route de Narbonne, 31062 Toulouse, France.},
author = {de Castro, Yohann},
journal = {Annales mathématiques Blaise Pascal},
keywords = {Isoperimetric inequalities; Asymmetry; Log-concave measures; Gaussian measure; isoperimetric inequalities; asymmetry; log-concave measures},
language = {eng},
month = {7},
number = {2},
pages = {251-271},
publisher = {Annales mathématiques Blaise Pascal},
title = {Quantitative Isoperimetric Inequalities on the Real Line},
url = {http://eudml.org/doc/219759},
volume = {18},
year = {2011},
}

TY - JOUR
AU - de Castro, Yohann
TI - Quantitative Isoperimetric Inequalities on the Real Line
JO - Annales mathématiques Blaise Pascal
DA - 2011/7//
PB - Annales mathématiques Blaise Pascal
VL - 18
IS - 2
SP - 251
EP - 271
AB - In a recent paper A. Cianchi, N. Fusco, F. Maggi, and A. Pratelli have shown that, in the Gauss space, a set of given measure and almost minimal Gauss boundary measure is necessarily close to be a half-space.Using only geometric tools, we extend their result to all symmetric log-concave measures on the real line. We give sharp quantitative isoperimetric inequalities and prove that among sets of given measure and given asymmetry (distance to half line, i.e. distance to sets of minimal perimeter), the intervals or complements of intervals have minimal perimeter.
LA - eng
KW - Isoperimetric inequalities; Asymmetry; Log-concave measures; Gaussian measure; isoperimetric inequalities; asymmetry; log-concave measures
UR - http://eudml.org/doc/219759
ER -

References

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  1. L. Ambrosio, N. Fusco, D. Pallara, Functions of bounded variation and free discontinuity problems, (2000), The Clarendon Press Oxford University Press, New York Zbl0957.49001MR1857292
  2. S. G. Bobkov, An isoperimetric problem on the line, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 216 (1994), 5-9, 161 Zbl0868.60014MR1327260
  3. C. Borell, The Brunn-Minkowski inequality in Gauss space, Invent. Math. 30 (1975), 207-216 Zbl0292.60004MR399402
  4. A. Cianchi, N. Fusco, F. Maggi, A. Pratelli, On the isoperimetric deficit in Gauss space, American Journal of Mathematics 133 (2011), 131-186 Zbl1219.28005MR2752937
  5. N. Fusco, F. Maggi, A. Pratelli, The sharp quantitative isoperimetric inequality, Ann. of Math. (2) 168 (2008), 941-980 Zbl1187.52009MR2456887
  6. V. N. Sudakov, B. S. Cirel{'}son, Extremal properties of half-spaces for spherically invariant measures, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 41 (1974), 14-24, 165 Zbl0351.28015MR365680

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