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On the Brunn-Minkowski inequality for general measures with applications to new isoperimetric-type inequalities


Authors: Galyna Livshyts, Arnaud Marsiglietti, Piotr Nayar and Artem Zvavitch
Journal: Trans. Amer. Math. Soc. 369 (2017), 8725-8742
MSC (2010): Primary 52A40; Secondary 60G15
DOI: https://doi.org/10.1090/tran/6928
Published electronically: April 11, 2017
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Abstract: In this paper we present new versions of the classical Brunn-Minkowski inequality for different classes of measures and sets. We show that the inequality

$\displaystyle \mu (\lambda A + (1-\lambda )B)^{1/n} \geq \lambda \mu (A)^{1/n} + (1-\lambda )\mu (B)^{1/n} $

holds true for an unconditional product measure $ \mu $ with non-increasing density and a pair of unconditional convex bodies $ A,B \subset \mathbb{R}^n$. We also show that the above inequality is true for any unconditional $ \log $-concave measure $ \mu $ and unconditional convex bodies $ A,B \subset \mathbb{R}^n$. Finally, we prove that the inequality is true for a symmetric $ \log $-concave measure $ \mu $ and a pair of symmetric convex sets $ A,B \subset \mathbb{R}^2$, which, in particular, settles the two-dimensional case of the conjecture for Gaussian measure proposed by Gardner and Zvavitch in 2010.

In addition, we note that in the cases when the above inequality is true, one can deduce from it the $ 1/n$-concavity of the parallel volume $ t \mapsto \mu (A+tB)$, Brunn's type theorem and certain analogues of Minkowski's first inequality.


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  • [1] K. Ball, Isometric problems in $ \ell _p$ and sections of convex sets, PhD Dissertation, Cambridge (1986).
  • [2] Franck Barthe and Nolwen Huet, On Gaussian Brunn-Minkowski inequalities, Studia Math. 191 (2009), no. 3, 283-304. MR 2481898, https://doi.org/10.4064/sm191-3-9
  • [3] C. Borell, Convex set functions in $ d$-space, Period. Math. Hungar. 6 (1975), no. 2, 111-136. MR 0404559
  • [4] Christer Borell, The Brunn-Minkowski inequality in Gauss space, Invent. Math. 30 (1975), no. 2, 207-216. MR 0399402
  • [5] Christer Borell, The Ehrhard inequality, C. R. Math. Acad. Sci. Paris 337 (2003), no. 10, 663-666 (English, with English and French summaries). MR 2030108, https://doi.org/10.1016/j.crma.2003.09.031
  • [6] Károly J. Böröczky, Erwin Lutwak, Deane Yang, and Gaoyong Zhang, The log-Brunn-Minkowski inequality, Adv. Math. 231 (2012), no. 3-4, 1974-1997. MR 2964630, https://doi.org/10.1016/j.aim.2012.07.015
  • [7] Károly J. Böröczky, Erwin Lutwak, Deane Yang, and Gaoyong Zhang, The logarithmic Minkowski problem, J. Amer. Math. Soc. 26 (2013), no. 3, 831-852. MR 3037788, https://doi.org/10.1090/S0894-0347-2012-00741-3
  • [8] D. Cordero-Erausquin, M. Fradelizi, and B. Maurey, The (B) conjecture for the Gaussian measure of dilates of symmetric convex sets and related problems, J. Funct. Anal. 214 (2004), no. 2, 410-427. MR 2083308, https://doi.org/10.1016/j.jfa.2003.12.001
  • [9] Max H. M. Costa and Thomas M. Cover, On the similarity of the entropy power inequality and the Brunn-Minkowski inequality, IEEE Trans. Inform. Theory 30 (1984), no. 6, 837-839. MR 782217, https://doi.org/10.1109/TIT.1984.1056983
  • [10] Antoine Ehrhard, Symétrisation dans l'espace de Gauss, Math. Scand. 53 (1983), no. 2, 281-301 (French). MR 745081, https://doi.org/10.7146/math.scand.a-12035
  • [11] Matthieu Fradelizi and Arnaud Marsiglietti, On the analogue of the concavity of entropy power in the Brunn-Minkowski theory, Adv. in Appl. Math. 57 (2014), 1-20. MR 3206519, https://doi.org/10.1016/j.aam.2014.02.004
  • [12] R. J. Gardner, The Brunn-Minkowski inequality, Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 3, 355-405. MR 1898210, https://doi.org/10.1090/S0273-0979-02-00941-2
  • [13] Richard J. Gardner, Geometric tomography, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 58, Cambridge University Press, New York, 2006. MR 2251886
  • [14] Richard J. Gardner and Artem Zvavitch, Gaussian Brunn-Minkowski inequalities, Trans. Amer. Math. Soc. 362 (2010), no. 10, 5333-5353. MR 2657682, https://doi.org/10.1090/S0002-9947-2010-04891-3
  • [15] Olivier Guédon, Piotr Nayar, and Tomasz Tkocz, Concentration inequalities and geometry of convex bodies, Analytical and probabilistic methods in the geometry of convex bodies, IMPAN Lect. Notes, vol. 2, Polish Acad. Sci. Inst. Math., Warsaw, 2014, pp. 9-86. MR 3329056
  • [16] R. Henstock and A. M. Macbeath, On the measure of sum-sets. I. The theorems of Brunn, Minkowski, and Lusternik, Proc. London Math. Soc. (3) 3 (1953), 182-194. MR 0056669
  • [17] Stanisław Kwapień and Jerzy Sawa, On some conjecture concerning Gaussian measures of dilatations of convex symmetric sets, Studia Math. 105 (1993), no. 2, 173-187. MR 1226627
  • [18] Rafał Latała, A note on the Ehrhard inequality, Studia Math. 118 (1996), no. 2, 169-174. MR 1389763
  • [19] R. Latała, On some inequalities for Gaussian measures, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 813-822. MR 1957087
  • [20] Rafał Latała and Krzysztof Oleszkiewicz, Gaussian measures of dilatations of convex symmetric sets, Ann. Probab. 27 (1999), no. 4, 1922-1938. MR 1742894, https://doi.org/10.1214/aop/1022677554
  • [21] Rafał Latała and Krzysztof Oleszkiewicz, Small ball probability estimates in terms of widths, Studia Math. 169 (2005), no. 3, 305-314. MR 2140804, https://doi.org/10.4064/sm169-3-6
  • [22] L. Leindler, On a certain converse of Hölder's inequality. II, Acta Sci. Math. (Szeged) 33 (1972), no. 3-4, 217-223. MR 2199372
  • [23] Amir Livne Bar-on, The (B) conjecture for uniform measures in the plane, Geometric aspects of functional analysis, Lecture Notes in Math., vol. 2116, Springer, Cham, 2014, pp. 341-353. MR 3364696, https://doi.org/10.1007/978-3-319-09477-9_22
  • [24] Arnaud Marsiglietti, Concavity properties of extensions of the parallel volume, Mathematika 62 (2016), no. 1, 266-282. MR 3430383, https://doi.org/10.1112/S0025579314000369
  • [25] Arnaud Marsiglietti, On the improvement of concavity of convex measures, Proc. Amer. Math. Soc. 144 (2016), no. 2, 775-786. MR 3430853, https://doi.org/10.1090/proc/12694
  • [26] Piotr Nayar and Tomasz Tkocz, A note on a Brunn-Minkowski inequality for the Gaussian measure, Proc. Amer. Math. Soc. 141 (2013), no. 11, 4027-4030. MR 3091793, https://doi.org/10.1090/S0002-9939-2013-11609-6
  • [27] Piotr Nayar and Tomasz Tkocz, S-inequality for certain product measures, Math. Nachr. 287 (2014), no. 4, 398-404. MR 3179670, https://doi.org/10.1002/mana.201200294
  • [28] Piotr Nayar and Tomasz Tkocz, The unconditional case of the complex $ S$-inequality, Israel J. Math. 197 (2013), no. 1, 99-106. MR 3096608, https://doi.org/10.1007/s11856-012-0178-x
  • [29] András Prékopa, On logarithmic concave measures and functions, Acta Sci. Math. (Szeged) 34 (1973), 335-343. MR 0404557
  • [30] Christos Saroglou, Remarks on the conjectured log-Brunn-Minkowski inequality, Geom. Dedicata 177 (2015), 353-365. MR 3370038, https://doi.org/10.1007/s10711-014-9993-z
  • [31] Christos Saroglou, More on logarithmic sums of convex bodies, Mathematika 62 (2016), no. 3, 818-841. MR 3521355, https://doi.org/10.1112/S0025579316000061
  • [32] Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Second expanded edition, Encyclopedia of Mathematics and its Applications, vol. 151, Cambridge University Press, Cambridge, 2014. MR 3155183
  • [33] V. N. Sudakov and B. S. Cirelson, Extremal properties of half-spaces for spherically invariant measures, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 41 (1974), 14-24, 165 (Russian). Problems in the theory of probability distributions, II. MR 0365680

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Additional Information

Galyna Livshyts
Affiliation: 228 Skiles building, School of sciences, Georgia Institute of Technology, 686 Cherry Street NW, Atlanta, Georgia 30332
Email: glivshyts6@math.gatech.edu

Arnaud Marsiglietti
Affiliation: Institute for Mathematics and Its Applications, University of Minnesota, 207 Church Street SE, 434 Lind Hall, Minneapolis, Minnesota 55455
Email: arnaud.marsiglietti@ima.umn.edu

Piotr Nayar
Affiliation: Wharton Department of Statistics, University of Pennsylvania, 432-1 Jon M. Huntsman Hall, 3730 Walnut Street, Philadelphia, Pennsylvania 19104
Email: nayar@mimuw.edu.pl

Artem Zvavitch
Affiliation: Department of Mathematical Sciences, Kent State University, Kent, Ohio 44242
Email: zvavitch@math.kent.edu

DOI: https://doi.org/10.1090/tran/6928
Keywords: Convex body, Gaussian measure, Brunn-Minkowski inequality, Minkowski first inequality, S-inequality, Brunn's theorem, Gaussian isoperimetry, log-Brunn-Minkowski inequality
Received by editor(s): July 9, 2015
Received by editor(s) in revised form: February 7, 2016, and February 25, 2016
Published electronically: April 11, 2017
Additional Notes: The first author was supported in part by the U.S. National Science Foundation Grant DMS-1101636
The second author was supported in part by the Institute for Mathematics and Its Applications with funds provided by the National Science Foundation
The third author was supported in part by NCN grant DEC-2012/05/B/ST1/00412
The fourth author was supported in part by the U.S. National Science Foundation Grant DMS-1101636 and the Simons Foundation
Article copyright: © Copyright 2017 American Mathematical Society

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