Abstract
The authors prove a quantitative stability result for the Brunn-Minkowski inequality on sets of equal volume: If |A| = |B| > 0 and |A + B|1/n = (2+δ)|A|1/n for some small δ, then, up to a translation, both A and B are close (in terms of δ) to a convex set K. Although this result was already proved by the authors in a previous paper, the present paper provides a more elementary proof that the authors believe has its own interest. Also, the result here provides a stronger estimate for the stability exponent than the previous result of the authors.
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References
Christ, M., Near equality in the two-dimensional Brunn-Minkowski inequality, Preprint, 2012. http://arxiv. org/abs/1206.1965
Christ, M., Near equality in the Brunn-Minkowski inequality, Preprint, 2012. http://arxiv.org/abs/1207.5062
Christ, M., An approximate inverse Riesz-Sobolev inequality, Preprint, 2011. http://arxiv.org/abs/1112.3715
Christ, M., Personal communication.
Diskant, V. I., Stability of the solution of a Minkowski equation (in Russian), Sibirsk. Mat. Z. 14, 1973, 669–673, 696.
Figalli, A., Stability results for the Brunn-Minkowski inequality, Colloquium De Giorgi 2013 and 2014, 119–127, Colloquia, 5, Ed. Norm., Pisa, 2014.
Figalli, A., Quantitative stability results for the Brunn-Minkowski inequality, Proceedings of the ICM 2014, to appear.
Figalli, A. and Jerison D., Quantitative stability for sumsets in Rn, J. Eur. Math. Soc. (JEMS), 17(5), 2015, 1079–1106.
Figalli, A. and Jerison, D., Quantitative stability for the Brunn-Minkowski inequality, Adv. Math., to appear.
Figalli, A., Maggi, F. and Pratelli, A., A mass transportation approach to quantitative isoperimetric inequalities, Invent. Math., 182(1), 2010, 167–211.
Figalli, A., Maggi, F. and Pratelli, A., A refined Brunn-Minkowski inequality for convex sets, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26(6), 2009, 2511–2519.
Freiman, G. A., The addition of finite sets. I (in Russian), Izv. Vyss. Ucebn. Zaved. Matematika, 13(6), 1959, 202–213.
Freiman, G. A., Foundations of a structural theory of set addition, Translated from the Russian, Translations of Mathematical Monographs, Vol. 37. American Mathematical Society, Providence, RI, 1973.
Gardner, R. J., The Brunn-Minkowski inequality, Bull. Amer. Math. Soc. (N.S.) 39(3), 2002, 355–405.
Groemer, H., On the Brunn-Minkowski theorem, Geom. Dedicata, 27(3), 1988, 357–371.
John F., Extremum problems with inequalities as subsidiary conditions, Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, 187–204; Interscience, New York, 1948.
Tao, T. and Vu, V., Additive combinatorics, Cambridge Studies in Advanced Mathematics, 105, Cambridge University Press, Cambridge, 2006.
Acknowledgments
This work started during Alessio Figalli’s visit at MIT during the fall 2012. Alessio Figalli wishes to thank the Mathematics Department at MIT for its warm hospitality.
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Dedicated to Professor Haim Brezis on the occasion of his 70th birthday
This work was supported by NSF Grant DMS-1262411, NSF Grant DMS-1361122, NSF Grant DMS-1069225 and DMS-1500771.
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Figalli, A., Jerison, D. Quantitative stability of the Brunn-Minkowski inequality for sets of equal volume. Chin. Ann. Math. Ser. B 38, 393–412 (2017). https://doi.org/10.1007/s11401-017-1075-8
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DOI: https://doi.org/10.1007/s11401-017-1075-8