A discrete version of the Brunn-Minkowski inequality and its stability

Michel Bonnefont[1]

  • [1] Institut de Mathématiques – UMR 5219 Université de Toulouse et CNRS 118 route de Narbonne 31062 Toulouse FRANCE

Annales mathématiques Blaise Pascal (2009)

  • Volume: 16, Issue: 2, page 245-257
  • ISSN: 1259-1734

Abstract

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In the first part of the paper, we define an approximated Brunn-Minkowski inequality which generalizes the classical one for metric measure spaces. Our new definition, based only on properties of the distance, allows also us to deal with discrete metric measure spaces. Then we show the stability of our new inequality under convergence of metric measure spaces. This result gives as corollary the stability of the classical Brunn-Minkowski inequality for geodesic spaces. The proof of this stability was done for related inequalities (curvature-dimension inequality, metric contraction property) but not for the Brunn-Minkowski one, as far as we know.In the second part of the paper, we show that every metric measure space satisfying the classical Brunn-Minkowski inequality can be approximated by discrete metric spaces with some approximated Brunn-Minkowski inequalities.

How to cite

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Bonnefont, Michel. "A discrete version of the Brunn-Minkowski inequality and its stability." Annales mathématiques Blaise Pascal 16.2 (2009): 245-257. <http://eudml.org/doc/10577>.

@article{Bonnefont2009,
abstract = {In the first part of the paper, we define an approximated Brunn-Minkowski inequality which generalizes the classical one for metric measure spaces. Our new definition, based only on properties of the distance, allows also us to deal with discrete metric measure spaces. Then we show the stability of our new inequality under convergence of metric measure spaces. This result gives as corollary the stability of the classical Brunn-Minkowski inequality for geodesic spaces. The proof of this stability was done for related inequalities (curvature-dimension inequality, metric contraction property) but not for the Brunn-Minkowski one, as far as we know.In the second part of the paper, we show that every metric measure space satisfying the classical Brunn-Minkowski inequality can be approximated by discrete metric spaces with some approximated Brunn-Minkowski inequalities.},
affiliation = {Institut de Mathématiques – UMR 5219 Université de Toulouse et CNRS 118 route de Narbonne 31062 Toulouse FRANCE},
author = {Bonnefont, Michel},
journal = {Annales mathématiques Blaise Pascal},
keywords = {Brunn-Minkowski inequality; metric measure spaces; $\mathbb\{D\}$-convergence; Ricci curvature; discretization; -convergence},
language = {eng},
month = {7},
number = {2},
pages = {245-257},
publisher = {Annales mathématiques Blaise Pascal},
title = {A discrete version of the Brunn-Minkowski inequality and its stability},
url = {http://eudml.org/doc/10577},
volume = {16},
year = {2009},
}

TY - JOUR
AU - Bonnefont, Michel
TI - A discrete version of the Brunn-Minkowski inequality and its stability
JO - Annales mathématiques Blaise Pascal
DA - 2009/7//
PB - Annales mathématiques Blaise Pascal
VL - 16
IS - 2
SP - 245
EP - 257
AB - In the first part of the paper, we define an approximated Brunn-Minkowski inequality which generalizes the classical one for metric measure spaces. Our new definition, based only on properties of the distance, allows also us to deal with discrete metric measure spaces. Then we show the stability of our new inequality under convergence of metric measure spaces. This result gives as corollary the stability of the classical Brunn-Minkowski inequality for geodesic spaces. The proof of this stability was done for related inequalities (curvature-dimension inequality, metric contraction property) but not for the Brunn-Minkowski one, as far as we know.In the second part of the paper, we show that every metric measure space satisfying the classical Brunn-Minkowski inequality can be approximated by discrete metric spaces with some approximated Brunn-Minkowski inequalities.
LA - eng
KW - Brunn-Minkowski inequality; metric measure spaces; $\mathbb{D}$-convergence; Ricci curvature; discretization; -convergence
UR - http://eudml.org/doc/10577
ER -

References

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  5. R. J. Gardner, The Brunn-Minkowski inequality, Bull. Amer. Math. Soc. (N.S.) 39 (2002), 355-405 (electronic) Zbl1019.26008MR1898210
  6. Nicolas Juillet, Geometric Inequalities and Generalised Ricci Bounds in Heisenberg Group Zbl1176.53053
  7. John Lott, Cédric Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. (2) 169 (2009), 903-991 Zbl1178.53038MR2480619
  8. Shin-ichi Ohta, On the measure contraction property of metric measure spaces, Comment. Math. Helv. 82 (2007), 805-828 Zbl1176.28016MR2341840
  9. Karl-Theodor Sturm, On the geometry of metric measure spaces. I, Acta Math. 196 (2006), 65-131 Zbl1105.53035MR2237206
  10. Karl-Theodor Sturm, On the geometry of metric measure spaces. II, Acta Math. 196 (2006), 133-177 Zbl1106.53032MR2237207
  11. Cédric Villani, Topics in optimal transportation, 58 (2003), American Mathematical Society, Providence, RI Zbl1106.90001MR1964483

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