Journal Menu
Volume 10, Issue 2 (2018), Pages [41] - [153]
VOLUME INEQUALITIES FOR GENERAL 
ZONOIDS OF EVEN ISOTROPIC MEASURES
[1] K. Ball, Volume ratios and a reverse isoperimetric inequality, Journal of the London Mathematical Society 44(2) (1991), 351-359.
DOI: https://doi.org/10.1112/jlms/s2-44.2.351
[2] K. Ball, Volumes of Sections of Cubes and Related Problems, Geometric Aspects of Functional Analysis (1987-88), Lecture Notes in Mathematics, Volume 1376, Springer, Berlin, 1989, pp. 251-260.
[3] K. Ball, Convex Geometry and Functional Analysis, Handbook of the Geometry of Banach Spaces, Volume 1, North-Holland,
DOI: https://doi.org/10.1016/S1874-5849(01)80006-1
[4] K. Ball, Shadows of convex bodies, Trans. Amer. Math. Soc. 327(2) (1991), 891-901.
DOI: https://doi.org/10.1090/S0002-9947-1991-1035998-3
[5] K. Ball, Logarithmically concave functions and sections of convex sets in 
Studia Math. 88(1) (1988), 69-84.
DOI: https://doi.org/10.4064/sm-88-1-69-84
[6] F. Barthe, A Continuous Version of the Brascamp-Lieb inequalities, Geometric Aspects of Functional Analysis, Lecture Notes in Math., Volume 1850, Springer, Berlin, 2004, pp. 53-63.
DOI: https://doi.org/10.1007/978-3-540-44489-3_6
[7] F. Barthe, Inégalités de Brascamp-Lieb et convexité, Comptes Rendus de l’Académie des Sciences: Series I Mathematics 324(8) (1997), 885-888.
DOI: https://doi.org/10.1016/S0764-4442(97)86963-7
[8] F. Barthe, On a reverse form of the Brascamp-Lieb inequality, Invent. Math. 134(2) (1998), 335-361.
DOI: https://doi.org/10.1007/s002220050267
[9] F. Barthe and D. Cordero-Erausquin, Invariances in variance estimates, Proc. Lond. Math. Soc. 106(1) (2013), 33-64.
DOI: https://doi.org/10.1112/plms/pds011
[10] F. Barthe, D. Cordero-Erausquin, M. Ledoux and B. Maurey, Correlation and Brascamp-Lieb inequalities for Markov semigroups, Int. Math. Res. Not. 10 (2011), 2177-2216.
DOI: https://doi.org/10.1093/imrn/rnq114
[11] J. Bennett, A. Carbery, M. Christ and T. Tao, The Brascamp-Lieb inequalities: Finiteness, structure and extremals, Geom. Funct. Anal. 17(5) (2008), 1343-1415.
DOI: https://doi.org/10.1007/s00039-007-0619-6
[12] J. Bennett, A. Carbery and T. Tao, On the multilinear restriction and Kakeya conjectures, Acta Math. 196(2) (2006), 261-302.
DOI: https://doi.org/10.1007/s11511-006-0006-4
[13] J. Bennett, A. Carbery and J. Wright, A nonlinear generalization of the Loomis-Whitney inequality and applications, Math. Res. Lett. 12(4) (2005), 443-457.
DOI: http://dx.doi.org/10.4310/MRL.2005.v12.n4.a1
[14] A. Bernig, The isoperimetrix in the dual Brunn-Minkowski theory, Adv. Math. 254 (2014), 1-14.
DOI: https://doi.org/10.1016/j.aim.2013.12.020
[15] K. J. Böröczky and D. Hug, Isotropic measures and stronger forms of the reverse isoperimetric inequality, Trans. Amer. Math. Soc. 369(10) (2017), 6987-7019.
DOI: https://doi.org/10.1090/tran/6857
[16] K. J. Böröczky, E. Lutwak, D. Yang and G. Zhang, Affine images of isotropic measures, J. Differential Geom. 99(3) (2015), 407-442.
DOI: https://doi.org/10.4310/jdg/1424880981
[17] Károly J. Böröczky, Ferenc Fodor and Daniel Hug, Strengthened volume inequalities for .gif){kind=small}
zonoids of even isotropic measures, Trans. Amer. Math. Soc. (appear).
DOI: https://doi.org/10.1090/tran/7299
[18] S. G. Bobkov and F. L. Nazarov, On Convex Bodies and Log-Concave Probability Measures with Unconditional Basis, Geometric Aspects of Functional Analysis, in: Lecture Notes in Math., Vol. 1807, Springer,
[19] B. Bollobas and A. Thomason, Projections of bodies and hereditary properties of hypergraphs, Bull. Lond. Math. Soc. 27(5) (1995), 417-424.
DOI: https://doi.org/10.1112/blms/27.5.417
[20] H. J. Brascamp and E. H. Lieb, Best constants in Young’s inequality, its converse, and its generalization to more than three functions, Adv. Math. 20(2) (1976), 151-173.
DOI: https://doi.org/10.1016/0001-8708(76)90184-5
[21] S. Campi and P. Gronchi, The 
centroid inequality, Adv. Math. 167(1) (2002), 128-141.
DOI: https://doi.org/10.1006/aima.2001.2036
[22] S. Campi and P. Gronchi, On the reverse centroid inequality, Mathematika 49(1-2) (2002), 1-11.
DOI: https://doi.org/10.1112/S0025579300016004
[23] S. Campi and P. Gronchi, Estimates of Loomis-Whitney type for intrinsic volumes, Adv. in Appl. Math. 47(3) (2011), 545-561.
DOI: https://doi.org/10.1016/j.aam.2011.01.001
[24] E. A. Carlen and D. Cordero-Erausquin, Subadditivity of the entropy and its relation to Brascamp-Lieb type inequalities, Geom. Funct. Anal. 19(2) (2009), 373-405.
DOI: https://doi.org/10.1007/s00039-009-0001-y
[25] W. J. Firey, p-Means of convex bodies, Math. Scand. 10 (1962), 17-24.
DOI: https://doi.org/10.7146/math.scand.a-10510
[26] M. Fradelizi, Hyperplane sections of convex bodies in isotropic position, Beiträge Algebra Geom. 40(1) (1999), 163-183.
[27] R. J. Gardner, Geometric Tomography, Second Edition, Encylopedia Math. Appl.,
[28] R. J. Gardner, The dual Brunn-Minkowski theory for bounded Borel sets: Dual affine quermassintegrals and inequalities, Adv. Math. 216(1) (2007), 358-386.
DOI: https://doi.org/10.1016/j.aim.2007.05.018
[29] R. J. Gardner, D. Hug and W. Weil, Operations between sets in geometry, J. Eur. Math. Soc. 15(6) (2013), 2297-2352.
DOI: https://doi.org/10.4171/JEMS/422
[30] R. J. Gardner, D. Hug, W. Weil and D. Ye, The dual Orlicz-Brunn-Minkowski theory, J. Math. Anal. Appl. 430(2) (2015), 810-829.
DOI: https://doi.org/10.1016/j.jmaa.2015.05.016
[31] A. Giannopoulos and M. Papadimitrakis, Isotropic surface area measures, Mathematika 46(1) (1999), 1-13.
DOI: https://doi.org/10.1112/S0025579300007518
[32] O. Guédon and E. Milman, Interpolating thin-shell and sharp large-deviation estimates for isotropic log-concave measures, Geom. Funct. Anal. 21(5) (2011), 1043-1068.
DOI: https://doi.org/10.1007/s00039-011-0136-5
[33] C. Haberl, .gif){kind=small}
intersection bodies, Adv. Math. 217(6) (2008), 2599-2624.
DOI: https://doi.org/10.1016/j.aim.2007.11.013
[34] C. Haberl, Star body valued valuations, Indiana Univ. Math. J. 58(5) (2009), 2253-2276.
DOI: https://doi.org/10.1512/iumj.2009.58.3685
[35] C. Haberl and M. Ludwig, A characterization of intersection bodies, Int. Math. Res. Not. (2006), Article ID 10548.
DOI: https://doi.org/10.1155/IMRN/2006/10548
[36] C. Haberl and F. E. Schuster, General affine isoperimetric inequalities, J. Differential Geom. 83(1) (2009), 1-26.
DOI: https://doi.org/10.4310/jdg/1253804349
[37] D. Hensley, Slicing convex bodies-bounds of slice area in terms of the body’s covariance, Proc. Amer. Math. Soc. 79(4) (1980), 619-625.
DOI: https://doi.org/10.1090/S0002-9939-1980-0572315-5
[38] F. John, Polar correspondence with respect to a convex region, Duke Math. J. 3(3) (1937), 355-369.
DOI: https://doi.org/10.1215/S0012-7094-37-00327-2
[39] F. John, Extremum Problems with Inequalities as Subsidiary Conditions, in: Studies and Essays Presented to R. Courant on his 60th Birthday, Jannuary 8, Interscience Publishers, Inc.,
[40] R. Kannan, L. Lovász and M. Simonovits, Isoperimetric problems for convex bodies and a localization lemma, Discrete Comput. Geom. 13(3-4) (1995), 541-559.
DOI: https://doi.org/10.1007/BF02574061
[41] B. Klartag, A Berry-Esseen type inequality for convex bodies with an unconditional basis, Probab. Theory Related Fields 145(1-2) (2009), 1-33.
DOI: https://doi.org/10.1007/s00440-008-0158-6
[42] B. Klartag, On nearly radial marginals of high-dimensional probability measures, J. Eur. Math. Soc. 12(3) (2010), 723-754.
DOI: https://doi.org/10.4171/JEMS/213
[43] J. C. Kuang, Applied Inequalities, 4th Edition,
[44] D. R. Lewis, Finite dimensional subspaces of .gif){kind=small}
Studia Math. 63(2) (1978), 207-212.
DOI: https://doi.org/10.4064/sm-63-2-207-212
[45] E. H. Lieb, Gaussian kernels have only Gaussian maximizers, Invent. Math. 102(1) (1990), 179-208.
DOI: https://doi.org/10.1007/BF01233426
[46] A. J. Li and Q. Z. Huang, The Loomis-Whitney inequality, Adv. in Appl. Math. 75 (2016), 94-115.
DOI: https://doi.org/10.1016/j.aam.2016.01.003
[47] A. J. Li, G. Wang and G. S. Leng, An extended Loomis-Whitney inequality for positive double John bases, Glasg. Math. J. 53(3) (2011), 451-462.
[48] A. J. Li, D. M. Xi and G. Y. Zhang, Volume inequalities of convex bodies from cosine transforms on Grassmann manifolds, Adv. Math. 304 (2017), 494-538.
DOI: https://doi.org/10.1016/j.aim.2016.09.007
[49] L. H. Loomis and H. Whiteny, An inequality related to the isoperimetric inequality, Bull. Amer. Math. Soc. 55(10) (1949), 961-962.
DOI: https://doi.org/10.1090/S0002-9904-1949-09320-5
[50] M. Ludwig, Intersection bodies and valuations, Amer. J. Math. 128(6) (2006), 1409-1428.
DOI: https://doi.org/10.1353/ajm.2006.0046
[51] E. Lutwak, D. Yang and G. Zhang, Volume inequalities for subspaces of .gif){kind=small}
J. Differential Geom. 68(1) (2004), 159-184.
DOI: https://doi.org/10.4310/jdg/1102536713
[52] E. Lutwak, D. Yang and G. Zhang, Volume inequalities for isotropic measures, Amer. J. Math. 129(6) (2007), 1711-1723.
DOI: https://doi.org/10.1353/ajm.2007.0038
[53] E. Lutwak, D. Yang and G. Zhang, .gif){kind=small}
John ellipsoids, Proc. London Math. Soc. 90(2) (2005), 497-520.
DOI: https://doi.org/10.1112/S0024611504014996
[54] E. Lutwak, The Brunn-Minkowski-Firey theory I: Mixed volumes and the Minkowski problem, J. Differential Geom. 38(1) (1993), 131-150.
DOI: https://doi.org/10.4310/jdg/1214454097
[55] E. Lutwak, The Brunn-Minkowski-Firey theory II: Affine and geominimal surface areas, Adv. Math. 118(2) (1996), 244-294.
DOI: https://doi.org/10.1006/aima.1996.0022
[56] E. Lutwak, Dual mixed volumes, Pacific J. Math. 58 (1975), 531-538.
DOI: https://doi.org/10.2140/pjm.1975.58.531
[57] E. Lutwak, Intersection bodies and dual mixed volume, Adv. Math. 71(2) (1988), 232-261.
DOI: https://doi.org/10.1016/0001-8708(88)90077-1
[58] E. Lutwak, Centroid bodies and dual mixed volumes, Proc. London Math. Soc. 60(2) (1990), 365-391.
DOI: https://doi.org/10.1112/plms/s3-60.2.365
[59] E. Lutwak, D. Yang and G. Zhang, affine isoperimetric inequalities, J. Differential Geom. 56(1) (2000), 111-132.
DOI: https://doi.org/10.4310/jdg/1090347527
[60] E. Lutwak, D. Yang and G. Zhang, Orlicz centroid bodies, J. Differential Geom. 84(2) (2010), 365-387.
DOI: https://doi.org/10.4310/jdg/1274707317
[61] E. Lutwak and G. Zhang, Blaschke-Santaló inequalities, J. Differential Geom. 47(1) (1997), 1-16.
DOI: https://doi.org/10.4310/jdg/1214460036
[62] T. Y. Ma, The minimal dual orlicz surface area, Taiwanese J. Math. 20(2) (2016), 287-309.
DOI: https://doi.org/10.11650/tjm.20.2016.6632
[63] V. D. Milman and A. Pajor, Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space, Geometric Aspects of Functional Analysis, (J. Lindenstrauss and V. D. Milman, Editors), Lecture Notes in Mathematics, Volume 1376, Springer, Berlin, 1989, pp. 64-104.
DOI: https://doi.org/10.1007/BFb0090049
[64] E. Milman, Dual mixed volumes and the slicing problem, Adv. Math. 207(2) (2006), 566-598.
DOI: https://doi.org/10.1016/j.aim.2005.09.008
[65] C. M. Petty, Centroid surfaces, Pacific J. Math. 11(4) (1961), 1535-1547.
DOI: https://doi.org/10.2140/pjm.1961.11.1535
[66] C. M. Petty, Projection bodies, in: Proc. Coll. Convexity,
[67] R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Second Expanded Edition,
[68] R. Schneider and W. Weil, Zonoids and related topics, in: P. M. Gruber and J. M. Wills (Editors), Convexity and its Applications, Birkhäuser,
[69] G. Zhang, Centered bodies and dual mixed volumes, Trans. Amer. Math. Soc. 345(2) (1994), 777-801.
DOI: https://doi.org/10.1090/S0002-9947-1994-1254193-9
[70] G. Zhang, The affine Sobolev inequality, J. Differential Geom. 53(1) (1999), 183-202.
DOI: https://doi.org/10.4310/jdg/1214425451
[71] B. Zhu, J. Zhou and W. Xu, Dual Orlicz-Brunn-Minkowski theory, Adv. Math. 264 (2014), 700-725.