Strengthened volume inequalities for $L_p$ zonoids of even isotropic measures
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- by Károly J. Böröczky, Ferenc Fodor and Daniel Hug PDF
- Trans. Amer. Math. Soc. 371 (2019), 505-548 Request permission
Abstract:
We strengthen the volume inequalities for $L_p$ zonoids of even isotropic measures and for their duals, which are originally due to Ball, Barthe, and Lutwak, Yang, and Zhang. The special case $p=\infty$ yields a stability version of the reverse isoperimetric inequality for centrally symmetric convex bodies. Adding to known inequalities and stability results for the reverse isoperimetric inequality of arbitrary convex bodies, we state a conjecture on volume inequalities for $L_p$ zonoids of general centered (non-symmetric) isotropic measures.
We achieve our main results by strengthening Barthe’s measure transportation proofs of the rank one case of the geometric Brascamp–Lieb and reverse Brascamp–Lieb inequalities by estimating the derivatives of the transportation maps for a special class of probability density functions. Based on our argument, we phrase a conjecture about a possible stability version of the Brascamp–Lieb and reverse Brascamp–Lieb inequalities.
We also establish some geometric properties of the distribution of general isotropic measures that are essential to our argument. In particular, we prove a measure theoretic analog of the Dvoretzky–Rogers lemma.
References
- Keith Ball, Volumes of sections of cubes and related problems, Geometric aspects of functional analysis (1987–88), Lecture Notes in Math., vol. 1376, Springer, Berlin, 1989, pp. 251–260. MR 1008726, DOI 10.1007/BFb0090058
- Keith Ball, Shadows of convex bodies, Trans. Amer. Math. Soc. 327 (1991), no. 2, 891–901. MR 1035998, DOI 10.1090/S0002-9947-1991-1035998-3
- Keith Ball, Volume ratios and a reverse isoperimetric inequality, J. London Math. Soc. (2) 44 (1991), no. 2, 351–359. MR 1136445, DOI 10.1112/jlms/s2-44.2.351
- Keith Ball, Convex geometry and functional analysis, Handbook of the geometry of Banach spaces, Vol. I, North-Holland, Amsterdam, 2001, pp. 161–194. MR 1863692, DOI 10.1016/S1874-5849(01)80006-1
- Franck Barthe, Inégalités de Brascamp-Lieb et convexité, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), no. 8, 885–888 (French, with English and French summaries). MR 1450443, DOI 10.1016/S0764-4442(97)86963-7
- Franck Barthe, On a reverse form of the Brascamp-Lieb inequality, Invent. Math. 134 (1998), no. 2, 335–361. MR 1650312, DOI 10.1007/s002220050267
- F. Barthe, A continuous version of the Brascamp-Lieb inequalities, Geometric aspects of functional analysis, Lecture Notes in Math., vol. 1850, Springer, Berlin, 2004, pp. 53–63. MR 2087150, DOI 10.1007/978-3-540-44489-3_{6}
- F. Barthe and D. Cordero-Erausquin, Invariances in variance estimates, Proc. Lond. Math. Soc. (3) 106 (2013), no. 1, 33–64. MR 3020738, DOI 10.1112/plms/pds011
- Franck Barthe, Dario Cordero-Erausquin, Michel Ledoux, and Bernard Maurey, Correlation and Brascamp-Lieb inequalities for Markov semigroups, Int. Math. Res. Not. IMRN 10 (2011), 2177–2216. MR 2806562, DOI 10.1093/imrn/rnq114
- Felix Behrend, Über einige Affininvarianten konvexer Bereiche, Math. Ann. 113 (1937), no. 1, 713–747 (German). MR 1513119, DOI 10.1007/BF01571662
- Jonathan Bennett, Anthony Carbery, Michael Christ, and Terence Tao, The Brascamp-Lieb inequalities: finiteness, structure and extremals, Geom. Funct. Anal. 17 (2008), no. 5, 1343–1415. MR 2377493, DOI 10.1007/s00039-007-0619-6
- Károly J. Böröczky and Martin Henk, Cone-volume measure of general centered convex bodies, Adv. Math. 286 (2016), 703–721. MR 3415694, DOI 10.1016/j.aim.2015.09.021
- Károly J. Böröczky and Daniel Hug, Isotropic measures and stronger forms of the reverse isoperimetric inequality, Trans. Amer. Math. Soc. 369 (2017), no. 10, 6987–7019. MR 3683100, DOI 10.1090/tran/6857
- Károly J. Böröczky, Erwin Lutwak, Deane Yang, and Gaoyong Zhang, Affine images of isotropic measures, J. Differential Geom. 99 (2015), no. 3, 407–442. MR 3316972
- Herm Jan Brascamp and Elliott H. Lieb, Best constants in Young’s inequality, its converse, and its generalization to more than three functions, Advances in Math. 20 (1976), no. 2, 151–173. MR 412366, DOI 10.1016/0001-8708(76)90184-5
- Silouanos Brazitikos, Apostolos Giannopoulos, Petros Valettas, and Beatrice-Helen Vritsiou, Geometry of isotropic convex bodies, Mathematical Surveys and Monographs, vol. 196, American Mathematical Society, Providence, RI, 2014. MR 3185453, DOI 10.1090/surv/196
- Eric A. Carlen and Dario Cordero-Erausquin, Subadditivity of the entropy and its relation to Brascamp-Lieb type inequalities, Geom. Funct. Anal. 19 (2009), no. 2, 373–405. MR 2545242, DOI 10.1007/s00039-009-0001-y
- L. Dalla, D. G. Larman, P. Mani-Levitska, and C. Zong, The blocking numbers of convex bodies, Discrete Comput. Geom. 24 (2000), no. 2-3, 267–277. The Branko Grünbaum birthday issue. MR 1758049, DOI 10.1007/s004540010032
- V. I. Diskant, Stability of the solution of a Minkowski equation, Sibirsk. Mat. Ž. 14 (1973), 669–673, 696 (Russian). MR 0333988
- L. Dümbgen, Bounding standard Gaussian tail probabilities, arXiv:1012.2063v3.
- A. Dvoretzky and C. A. Rogers, Absolute and unconditional convergence in normed linear spaces, Proc. Nat. Acad. Sci. U.S.A. 36 (1950), 192–197. MR 33975, DOI 10.1073/pnas.36.3.192
- László Fejes Tóth, Lagerungen in der Ebene auf der Kugel und im Raum, Die Grundlehren der mathematischen Wissenschaften, Band 65, Springer-Verlag, Berlin-New York, 1972 (German). Zweite verbesserte und erweiterte Auflage. MR 0353117
- A. Figalli, F. Maggi, and A. Pratelli, A refined Brunn-Minkowski inequality for convex sets, Ann. Inst. H. Poincaré C Anal. Non Linéaire 26 (2009), no. 6, 2511–2519. MR 2569906, DOI 10.1016/j.anihpc.2009.07.004
- A. Figalli, F. Maggi, and A. Pratelli, A mass transportation approach to quantitative isoperimetric inequalities, Invent. Math. 182 (2010), no. 1, 167–211. MR 2672283, DOI 10.1007/s00222-010-0261-z
- N. Fusco, F. Maggi, and A. Pratelli, The sharp quantitative isoperimetric inequality, Ann. of Math. (2) 168 (2008), no. 3, 941–980. MR 2456887, DOI 10.4007/annals.2008.168.941
- A. Giannopoulos and M. Papadimitrakis, Isotropic surface area measures, Mathematika 46 (1999), no. 1, 1–13. MR 1750398, DOI 10.1112/S0025579300007518
- H. Groemer, Stability properties of geometric inequalities, Amer. Math. Monthly 97 (1990), no. 5, 382–394. MR 1048910, DOI 10.2307/2324388
- H. Groemer, Stability of geometric inequalities, Handbook of convex geometry, Vol. A, B, North-Holland, Amsterdam, 1993, pp. 125–150. MR 1242978
- H. Groemer and R. Schneider, Stability estimates for some geometric inequalities, Bull. London Math. Soc. 23 (1991), no. 1, 67–74. MR 1111537, DOI 10.1112/blms/23.1.67
- Peter M. Gruber, Convex and discrete geometry, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 336, Springer, Berlin, 2007. MR 2335496
- Peter M. Gruber and Franz E. Schuster, An arithmetic proof of John’s ellipsoid theorem, Arch. Math. (Basel) 85 (2005), no. 1, 82–88. MR 2155113, DOI 10.1007/s00013-005-1326-x
- Olivier Guédon and Emanuel Milman, Interpolating thin-shell and sharp large-deviation estimates for isotropic log-concave measures, Geom. Funct. Anal. 21 (2011), no. 5, 1043–1068. MR 2846382, DOI 10.1007/s00039-011-0136-5
- Fritz John, Polar correspondence with respect to a convex region, Duke Math. J. 3 (1937), no. 2, 355–369. MR 1545993, DOI 10.1215/S0012-7094-37-00327-2
- R. Kannan, L. Lovász, and M. Simonovits, Isoperimetric problems for convex bodies and a localization lemma, Discrete Comput. Geom. 13 (1995), no. 3-4, 541–559. MR 1318794, DOI 10.1007/BF02574061
- Bo’az Klartag, A Berry-Esseen type inequality for convex bodies with an unconditional basis, Probab. Theory Related Fields 145 (2009), no. 1-2, 1–33. MR 2520120, DOI 10.1007/s00440-008-0158-6
- Bo’az Klartag, On nearly radial marginals of high-dimensional probability measures, J. Eur. Math. Soc. (JEMS) 12 (2010), no. 3, 723–754. MR 2639317, DOI 10.4171/JEMS/213
- D. R. Lewis, Finite dimensional subspaces of $L_{p}$, Studia Math. 63 (1978), no. 2, 207–212. MR 511305, DOI 10.4064/sm-63-2-207-212
- Elliott H. Lieb, Gaussian kernels have only Gaussian maximizers, Invent. Math. 102 (1990), no. 1, 179–208. MR 1069246, DOI 10.1007/BF01233426
- Erwin Lutwak, Selected affine isoperimetric inequalities, Handbook of convex geometry, Vol. A, B, North-Holland, Amsterdam, 1993, pp. 151–176. MR 1242979, DOI 10.1016/B978-0-444-89596-7.50010-9
- Erwin Lutwak, Deane Yang, and Gaoyong Zhang, Volume inequalities for subspaces of $L_p$, J. Differential Geom. 68 (2004), no. 1, 159–184. MR 2152912
- Erwin Lutwak, Deane Yang, and Gaoyong Zhang, $L_p$ John ellipsoids, Proc. London Math. Soc. (3) 90 (2005), no. 2, 497–520. MR 2142136, DOI 10.1112/S0024611504014996
- Erwin Lutwak, Deane Yang, and Gaoyong Zhang, Volume inequalities for isotropic measures, Amer. J. Math. 129 (2007), no. 6, 1711–1723. MR 2369894, DOI 10.1353/ajm.2007.0038
- Erwin Lutwak, Deane Yang, and Gaoyong Zhang, A volume inequality for polar bodies, J. Differential Geom. 84 (2010), no. 1, 163–178. MR 2629512
- Márton Naszódi, Proof of a conjecture of Bárány, Katchalski and Pach, Discrete Comput. Geom. 55 (2016), no. 1, 243–248. MR 3439267, DOI 10.1007/s00454-015-9753-3
- C. M. Petty, Surface area of a convex body under affine transformations, Proc. Amer. Math. Soc. 12 (1961), 824–828. MR 130618, DOI 10.1090/S0002-9939-1961-0130618-0
- 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
- Manuel Weberndorfer, Shadow systems of asymmetric $L_p$ zonotopes, Adv. Math. 240 (2013), 613–635. MR 3046320, DOI 10.1016/j.aim.2013.02.022
- J. G. Wendel, Note on the gamma function, Amer. Math. Monthly 55 (1948), 563–564. MR 29448, DOI 10.2307/2304460
Additional Information
- Károly J. Böröczky
- Affiliation: MTA Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda u. 13-15, 1053 Budapest, Hungary
- Email: carlos@renyi.hu
- Ferenc Fodor
- Affiliation: Department of Geometry, Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, 6720 Szeged, Hungary
- MR Author ID: 619845
- Email: fodorf@math.u-szeged.hu
- Daniel Hug
- Affiliation: Karlsruhe Institute of Technology (KIT), D-76128 Karlsruhe, Germany
- MR Author ID: 363423
- Email: daniel.hug@kit.edu
- Received by editor(s): September 2, 2016
- Received by editor(s) in revised form: April 19, 2017
- Published electronically: July 20, 2018
- Additional Notes: The first and second authors were supported by National Research, Development and Innovation Office – NKFIH grant 116451, and the first author was also supported by grant 109789.
The second author wishes to thank the MTA Alfréd Rényi Institute of Mathematics, where part of this work was done while he was a visiting researcher.
The third author was supported by DFG grants FOR 1548 and HU 1874/4-2. - © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 505-548
- MSC (2010): Primary 52A40; Secondary 52A38, 52B12, 26D15
- DOI: https://doi.org/10.1090/tran/7299
- MathSciNet review: 3885153