Abstract
A new family of ellipsoids associated with a pair of convex bodies in ℝn is introduced, which extends the classical John ellipsoids and the evolved Lp John ellipsoids in a distinctive way. Analogues of John’s inclusion and Ball’s volume-ratio inequality are established for these new ellipsoids.
Communicated by: M. Henk
Funding: Research of the authors is supported by NSFC No. 11471206 and 11871373.
References
[1] K. Ball, Volumes of sections of cubes and related problems. In: Geometric aspects of functional analysis (1987–88), volume 1376 of Lecture Notes in Math., 251–260, Springer 1989. MR1008726 Zbl 0674.4600810.1007/BFb0090058Suche in Google Scholar
[2] K. Ball, Volume ratios and a reverse isoperimetric inequality. J. London Math. Soc. (2) 44 (1991), 351–359. MR1136445 Zbl 0694.4601010.1112/jlms/s2-44.2.351Suche in Google Scholar
[3] K. Ball, Ellipsoids of maximal volume in convex bodies. Geom. Dedicata41 (1992), 241–250. MR1153987 Zbl 0747.5200710.1007/BF00182424Suche in Google Scholar
[4] F. Barthe, On a reverse form of the Brascamp-Lieb inequality. Invent. Math. 134 (1998), 335–361. MR1650312 Zbl 0901.2601010.1007/s002220050267Suche in Google Scholar
[5] J. Bastero, M. Romance, Positions of convex bodies associated to extremal problems and isotropic measures. Adv. Math. 184 (2004), 64–88. MR2047849 Zbl 1053.5201110.1016/S0001-8708(03)00095-1Suche in Google Scholar
[6] K. J. Böröczky, E. Lutwak, D. Yang, G. Zhang, The logarithmic Minkowski problem. J. Amer. Math. Soc. 26 (2013), 831–852. MR3037788 Zbl 1272.5201210.1090/S0894-0347-2012-00741-3Suche in Google Scholar
[7] S. Campi, P. Gronchi, The Lp-Busemann-Petty centroid inequality. Adv. Math. 167 (2002), 128–141. MR1901248 Zbl 1002.5200510.1006/aima.2001.2036Suche in Google Scholar
[8] K.-S. Chou, X.-J. Wang, The Lp-Minkowski problem and the Minkowski problem in centroaffine geometry. Adv. Math. 205 (2006), 33–83. MR2254308 Zbl 1245.5200110.1016/j.aim.2005.07.004Suche in Google Scholar
[9] B. Fleury, O. Guédon, G. Paouris, A stability result for mean width of Lp-centroid bodies. Adv. Math. 214 (2007), 865–877. MR2349721 Zbl 1132.5201210.1016/j.aim.2007.03.008Suche in Google Scholar
[10] R. J. Gardner, Geometric tomography, volume 58 of Encyclopedia of Mathematics and its Applications. Cambridge Univ. Press 2006. MR2251886 Zbl 1102.52002Suche in Google Scholar
[11] R. J. Gardner, D. Hug, W. Weil, Operations between sets in geometry. J. Eur. Math. Soc. 15 (2013), 2297–2352. MR3120744 Zbl 1282.5200610.4171/JEMS/422Suche in Google Scholar
[12] R. J. Gardner, D. Hug, W. Weil, The Orlicz–Brunn–Minkowski theory: a general framework, additions, and inequalities. J. Differential Geom. 97 (2014), 427–476. MR3263511 Zbl 1303.5200210.4310/jdg/1406033976Suche in Google Scholar
[13] A. Giannopoulos, M. Papadimitrakis, Isotropic surface area measures. Mathematika46 (1999), 1–13. MR1750398 Zbl 0960.5200910.1112/S0025579300007518Suche in Google Scholar
[14] A. A. Giannopoulos, V. D. Milman, Extremal problems and isotropic positions of convex bodies. Israel J. Math. 117 (2000), 29–60. MR1760584 Zbl 0964.5200410.1007/BF02773562Suche in Google Scholar
[15] P. M. Gruber, Convex and discrete geometry. Springer 2007. MR2335496 Zbl 1139.52001Suche in Google Scholar
[16] P. M. Gruber, John and Loewner ellipsoids. Discrete Comput. Geom. 46 (2011), 776–788. MR2846178 Zbl 1241.5200210.1007/s00454-011-9354-8Suche in Google Scholar
[17] P. M. Gruber, F. E. Schuster, An arithmetic proof of John’s ellipsoid theorem. Arch. Math. (Basel) 85 (2005), 82–88. MR2155113 Zbl 1086.5200210.1007/s00013-005-1326-xSuche in Google Scholar
[18] C. Haberl, E. Lutwak, D. Yang, G. Zhang, The even Orlicz Minkowski problem. Adv. Math. 224 (2010), 2485–2510. MR2652213 Zbl 1198.5200310.1016/j.aim.2010.02.006Suche in Google Scholar
[19] C. Haberl, F. E. Schuster, General Lp affine isoperimetric inequalities. J. Differential Geom. 83 (2009), 1–26. MR2545028 Zbl 1185.5200510.4310/jdg/1253804349Suche in Google Scholar
[20] J. Hu, G. Xiong, A new affine invariant geometric functional for polytopes and its associated affine isoperimetric inequalities. Accepted by Int. Math. Res. Not. IMRN, 2018.10.1093/imrn/rnz090Suche in Google Scholar
[21] F. John, Extremum problems with inequalities as subsidiary conditions. In: Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, 187–204, Interscience Publ. 1948. MR0030135 Zbl 0034.10503Suche in Google Scholar
[22] B. Klartag, On John-type ellipsoids. In: Geometric aspects of functional analysis, volume 1850 of Lecture Notes in Math., 149–158, Springer 2004. MR2087157 Zbl 1067.5200410.1007/978-3-540-44489-3_14Suche in Google Scholar
[23] D. R. Lewis, Ellipsoids defined by Banach ideal norms. Mathematika26 (1979), 18–29. MR557122 Zbl 0438.4600610.1112/S0025579300009566Suche in Google Scholar
[24] M. Ludwig, Ellipsoids and matrix-valued valuations. Duke Math. J. 119 (2003), 159–188. MR1991649 Zbl 1033.5201210.1215/S0012-7094-03-11915-8Suche in Google Scholar
[25] M. Ludwig, General affine surface areas. Adv. Math. 224 (2010), 2346–2360. MR2652209 Zbl 1198.5200410.1016/j.aim.2010.02.004Suche in Google Scholar
[26] M. Ludwig, M. Reitzner, A classification of SL(n) invariant valuations. Ann. of Math. (2) 172 (2010), 1219–1267. MR2680490 Zbl 1223.5200710.4007/annals.2010.172.1219Suche in Google Scholar
[27] E. Lutwak, The Brunn–Minkowski–Firey theory. I. Mixed volumes and the Minkowski problem. J. Differential Geom. 38 (1993), 131–150. MR1231704 Zbl 0788.5200710.4310/jdg/1214454097Suche in Google Scholar
[28] E. Lutwak, The Brunn–Minkowski–Firey theory. II. Affine and geominimal surface areas. Adv. Math. 118 (1996), 244–294. MR1378681 Zbl 0853.5200510.1006/aima.1996.0022Suche in Google Scholar
[29] E. Lutwak, D. Yang, G. Zhang, Lp affine isoperimetric inequalities. J. Differential Geom. 56 (2000), 111–132. MR1863023 Zbl 1034.5200910.4310/jdg/1090347527Suche in Google Scholar
[30] E. Lutwak, D. Yang, G. Zhang, A new ellipsoid associated with convex bodies. Duke Math. J. 104 (2000), 375–390. MR1781476 Zbl 0974.5200810.1215/S0012-7094-00-10432-2Suche in Google Scholar
[31] E. Lutwak, D. Yang, G. Zhang, On the Lp-Minkowski problem. Trans. Amer. Math. Soc. 356 (2004), 4359–4370. MR2067123 Zbl 1069.5201010.1090/S0002-9947-03-03403-2Suche in Google Scholar
[32] E. Lutwak, D. Yang, G. Zhang, Lp John ellipsoids. Proc. London Math. Soc. (3) 90 (2005), 497–520. MR2142136 Zbl 1074.5200510.1112/S0024611504014996Suche in Google Scholar
[33] E. Lutwak, D. Yang, G. Zhang, Orlicz centroid bodies. J. Differential Geom. 84 (2010), 365–387. MR2652465 Zbl 1206.4905010.4310/jdg/1274707317Suche in Google Scholar
[34] E. Lutwak, D. Yang, G. Zhang, Orlicz projection bodies. Adv. Math. 223 (2010), 220–242. MR2563216 Zbl 0564396210.1016/j.aim.2009.08.002Suche in Google Scholar
[35] V. D. Milman, A. Pajor, Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space. In: Geometric aspects of functional analysis (1987–88), volume 1376 of Lecture Notes in Math., 64–104, Springer 1989. MR1008717 Zbl 0679.4601210.1007/BFb0090049Suche in Google Scholar
[36] G. Paouris, Concentration of mass on convex bodies. Geom. Funct. Anal. 16 (2006), 1021–1049. MR2276533 Zbl 1114.5200410.1007/s00039-006-0584-5Suche in Google Scholar
[37] C. M. Petty, Surface area of a convex body under affine transformations. Proc. Amer. Math. Soc. 12 (1961), 824–828. MR0130618 Zbl 0101.4030410.1090/S0002-9939-1961-0130618-0Suche in Google Scholar
[38] G. Pisier, The volume of convex bodies and Banach space geometry, volume 94 of Cambridge Tracts in Mathematics. Cambridge Univ. Press 1989. MR1036275 Zbl 0698.4600810.1017/CBO9780511662454Suche in Google Scholar
[39] R. Schneider, Convex bodies: the Brunn–Minkowski theory, volume 151 of Encyclopedia of Mathematics and its Applications. Cambridge Univ. Press 2014. MR3155183 Zbl 1287.52001Suche in Google Scholar
[40] F. E. Schuster, M. Weberndorfer, Volume inequalities for asymmetric Wulff shapes. J. Differential Geom. 92 (2012), 263–283. MR2998673 Zbl 1264.5301010.4310/jdg/1352297808Suche in Google Scholar
[41] C. Schütt, E. Werner, Surface bodies and p-affine surface area. Adv. Math. 187 (2004), 98–145. MR2074173 Zbl 1089.5200210.1016/j.aim.2003.07.018Suche in Google Scholar
[42] E. Werner, D. Ye, New Lp affine isoperimetric inequalities. Adv. Math. 218 (2008), 762–780. MR2414321 Zbl 1155.5200210.1016/j.aim.2008.02.002Suche in Google Scholar
[43] E. M. Werner, Rényi divergence and Lp-affine surface area for convex bodies. Adv. Math. 230 (2012), 1040–1059. MR2921171 Zbl 1251.5200310.1016/j.aim.2012.03.015Suche in Google Scholar
[44] V. Yaskin, M. Yaskina, Centroid bodies and comparison of volumes. Indiana Univ. Math. J. 55 (2006), 1175–1194. MR2244603 Zbl 1102.5200510.1512/iumj.2006.55.2761Suche in Google Scholar
[45] D. Zou, G. Xiong, Orlicz–John ellipsoids. Adv. Math. 265 (2014), 132–168. MR3255458 Zbl 1301.5201510.1016/j.aim.2014.07.034Suche in Google Scholar
[46] D. Zou, G. Xiong, Orlicz–Legendre ellipsoids. J. Geom. Anal. 26 (2016), 2474–2502. MR3511485 Zbl 1369.5201510.1007/s12220-015-9636-0Suche in Google Scholar
© 2019 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- A remark on the mixed scalar curvature of a manifold with two orthogonal totally umbilical distributions
- On mixed Lp John ellipsoids
- Valuations on convex functions and convex sets and Monge–Ampère operators
- Weierstrass points on Kummer extensions
- The Bonnet problem for harmonic maps to the three-sphere
- Maximal arcs in projective planes of order 16 and related designs
- On the decomposition of the small diagonal of a K3 surface
- Doubly transitive dimensional dual hyperovals: universal covers and non-bilinear examples
- Higgs bundles and fundamental group schemes
- A relation between the curvature ellipse and the curvature parabola
- Non-orientable three-submanifolds of G2-manifolds
- The growth of the first non-Euclidean filling volume function of the quaternionic Heisenberg group
- A class of analytic pairs of conjugate functions in dimension three
Artikel in diesem Heft
- Frontmatter
- A remark on the mixed scalar curvature of a manifold with two orthogonal totally umbilical distributions
- On mixed Lp John ellipsoids
- Valuations on convex functions and convex sets and Monge–Ampère operators
- Weierstrass points on Kummer extensions
- The Bonnet problem for harmonic maps to the three-sphere
- Maximal arcs in projective planes of order 16 and related designs
- On the decomposition of the small diagonal of a K3 surface
- Doubly transitive dimensional dual hyperovals: universal covers and non-bilinear examples
- Higgs bundles and fundamental group schemes
- A relation between the curvature ellipse and the curvature parabola
- Non-orientable three-submanifolds of G2-manifolds
- The growth of the first non-Euclidean filling volume function of the quaternionic Heisenberg group
- A class of analytic pairs of conjugate functions in dimension three