Czechoslovak Mathematical Journal, Vol. 67, No. 3, pp. 779-793, 2017

Some inequalities for radial Blaschke-Minkowski homomorphisms

Lewen Ji, Zhenbing Zeng

Received April 12, 2016.   First published August 9, 2017.

Lewen Ji (corresponding author), Zhenbing Zeng, Department of Mathematics, Shanghai University, 99 Shangda Rd., Baoshan Qu, Shanghai 200444, China, e-mail: jilewen2008@163.com, zbzeng@shu.edu.cn

Abstract: We establish some Brunn-Minkowski type inequalities for radial Blaschke-Minkowski homomorphisms with respect to Orlicz radial sums and differences of dual quermassintegrals.

Keywords: radial Blaschke-Minkowski homomorphism; Orlicz radial sum

Classification (MSC 2010): 52A20, 52A40

DOI: 10.21136/CMJ.2017.0180-16

Full text available as PDF.


References:
  [1] J. Abardia, A. Bernig: Projection bodies in complex vector spaces. Adv. Math. 227 (2011), 830-846. DOI 10.1016/j.aim.2011.02.013 | MR 2793024 | Zbl 1217.52009
  [2] S. Alesker, A. Bernig, F. E. Schuster: Harmonic analysis of translation invariant valuations. Geom. Funct. Anal. 21 (2011), 751-773. DOI 10.1007/s00039-011-0125-8 | MR 2827009 | Zbl 1228.53088
  [3] E. F. Beckenbach, R. Bellman: Inequalities. Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge, Band 30, Springer, Berlin (1965). MR 0192009 | Zbl 0126.28002
  [4] R. J. Gardner: A positive answer to the Busemann-Petty problem in three dimensions. Ann. Math. (2) 140 (1994), 435-447. DOI 10.2307/2118606 | MR 1298719 | Zbl 0826.52010
  [5] R. J. Gardner: Intersection bodies and the Busemann-Petty problem. Trans. Am. Math. Soc. 342 (1994), 435-445. DOI 10.2307/2154703 | MR 1201126 | Zbl 0801.52005
  [6] R. J. Gardner: The Brunn-Minkowski inequality. Bull. Am. Math. Soc., New Ser. 39 (2002), 355-405. DOI 10.1090/S0273-0979-02-00941-2 | MR 1898210 | Zbl 1019.26008
  [7] R. J. Gardner, D. Hug, W. Weil: The Orlicz-Brunn-Minkowski theory: a general framework, additions, and inequalities. J. Differ. Geom. 97 (2014), 427-476. DOI 10.4310/jdg/1406033976 | MR 3263511 | Zbl 1303.52002
  [8] R. J. Gardner, D. Hug, W. Weil, D. Ye: The dual Orlicz-Brunn-Minkowski theory. J. Math. Anal. Appl. 430 (2015), 810-829. DOI 10.1016/j.jmaa.2015.05.016 | MR 3351982 | Zbl 1320.52008
  [9] R. J. Gardner, A. Koldobsky, T. Schlumprecht: An analytic solution to the Busemann-Petty problem on sections of convex bodies. Ann. Math. (2) 149 (1999), 691-703. DOI 10.2307/120978 | MR 1689343 | Zbl 0937.52003
  [10] R. J. Gardner, L. Parapatits, F. E. Schuster: A characterization of Blaschke addition. Adv. Math. 254 (2014), 396-418. DOI 10.1016/j.aim.2013.11.017 | MR 3161103 | Zbl 1291.52011
  [11] C. Haberl: Star body valued valuations. Indiana Univ. Math. J. 58 (2009), 2253-2276. DOI 10.1512/iumj.2009.58.3685 | MR 2583498 | Zbl 1183.52003
  [12] A. Koldobsky: Intersection bodies, positive definite distributions, and the Busemann-Petty problem. Am. J. Math. 120 (1998), 827-840. DOI 10.1353/ajm.1998.0030 | MR 1637955 | Zbl 0914.52001
  [13] A. Koldobsky: Stability in the Busemann-Petty and Shephard problems. Adv. Math. 228 (2011), 2145-2161. DOI 10.1016/j.aim.2011.06.031 | MR 2836117 | Zbl 1228.52006
  [14] A. Koldobsky, D. Ma: Stability and slicing inequalities for intersection bodies. Geom. Dedicata 162 (2013), 325-335. DOI 10.1007/s10711-012-9729-x | MR 3009547 | Zbl 1261.52003
  [15] G. Leng: The Brunn-Minkowski inequality for volume differences. Adv. Appl. Math. 32 (2004), 615-624. DOI 10.1016/S0196-8858(03)00095-2 | MR 2042686 | Zbl 1056.52003
  [16] M. Ludwig: Projection bodies and valuations. Adv. Math. 172 (2002), 158-168. DOI 10.1016/S0001-8708(02)00021-X | MR 1942402 | Zbl 1019.52003
  [17] M. Ludwig: Intersection bodies and valuations. Am. J. Math. 128 (2006), 1409-1428. DOI 10.1353/ajm.2006.0046 | MR 2275906 | Zbl 1115.52007
  [18] E. Lutwak: Dual mixed volumes. Pac. J. Math. 58 (1975), 531-538. DOI 10.2140/pjm.1975.58.531 | MR 0380631 | Zbl 0273.52007
  [19] E. Lutwak: Intersection bodies and dual mixed volumes. Adv. Math. 71 (1988), 232-261. DOI 10.1016/0001-8708(88)90077-1 | MR 0963487 | Zbl 0657.52002
  [20] E. Lutwak: Inequalities for mixed projection bodies. Trans. Am. Math. Soc. 339 (1993), 901-916. DOI 10.2307/2154305 | MR 1124171 | Zbl 0784.52009
  [21] E. Lutwak: The Brunn-Minkowski-Firey theory I. Mixed volumes and the Minkowski problem. J. Differ. Geom. 38 (1993), 131-150. DOI 10.4310/jdg/1214454097 | MR 1231704 | Zbl 0788.52007
  [22] E. Lutwak, D. Yang, G. Zhang: Orlicz centroid bodies. J. Differ. Geom. 84 (2010), 365-387. DOI 10.4310/jdg/1274707317 | MR 2652465 | Zbl 1206.49050
  [23] E. Lutwak, D. Yang, G. Zhang: Orlicz projection bodies. Adv. Math. 223 (2010), 220-242. DOI 10.1016/j.aim.2009.08.002 | MR 2563216 | Zbl 05643962
  [24] R. Schneider: Convex Bodies: The Brunn-Minkowski Theory. Encyclopedia of Mathematics and Its Applications 151, Cambridge University Press, Cambridge (2014). DOI 10.1017/CBO9781139003858 | MR 3155183 | Zbl 1287.52001
  [25] F. E. Schuster: Volume inequalities and additive maps of convex bodies. Mathematika 53 (2006), 211-234. DOI 10.1112/S0025579300000103 | MR 2343256 | Zbl 1129.52002
  [26] F. E. Schuster: Valuations and Busemann-Petty type problems. Adv. Math. 219 (2008), 344-368. DOI 10.1016/j.aim.2008.05.001 | MR 2435426 | Zbl 1146.52003
  [27] W. Wang: $L_p$ Brunn-Minkowski type inequalities for Blaschke-Minkowski homomorphisms. Geom. Dedicata 164 (2013), 273-285. DOI 10.1007/s10711-012-9772-7 | MR 3054628 | Zbl 1280.52007
  [28] D. Xi, H. Jin, G. Leng: The Orlicz Brunn-Minkowski inequality. Adv. Math. 260 (2014), 350-374. DOI 10.1016/j.aim.2014.02.036 | MR 3209355 | Zbl 06298949
  [29] G. Xiong, D. Zou: Orlicz mixed quermassintegrals. Sci. China, Math. 57 (2014), 2549-2562. DOI 10.1007/s11425-014-4812-4 | MR 3275405 | Zbl 1328.52003
  [30] C.-J. Zhao: On radial Blaschke-Minkowski homomorphisms. Geom. Dedicata 167 (2013), 1-10. DOI 10.1007/s10711-012-9798-x | MR 3128767 | Zbl 1287.52009
  [31] C.-J. Zhao: Orlicz dual mixed volumes. Result. Math. 68 (2015), 93-104. DOI 10.1007/s00025-014-0424-0 | MR 3391494 | Zbl 1329.52008
  [32] C. Zhao, W.-S. Cheung: Radial Blaschke-Minkowski homomorphisms and volume differences. Geom. Dedicata 154 (2011), 81-91. DOI 10.1007/s10711-010-9568-6 | MR 2832712 | Zbl 1230.52023
  [33] C.-J. Zhao, G. Leng: Brunn-Minkowski inequality for mixed intersection bodies. J. Math. Anal. Appl. 301 (2005), 115-123. DOI 10.1016/j.jmaa.2004.07.013 | MR 2105924 | Zbl 1065.52006
  [34] B. Zhu, J. Zhou, W. Xu: Dual Orlicz-Brunn-Minkowski theory. Adv. Math. 264 (2014), 700-725. DOI 10.1016/j.aim.2014.07.019 | MR 3250296 | Zbl 1307.52004


Access to the full text of journal articles on this site is restricted to the subscribers of Myris Trade. To activate your access, please contact Myris Trade at myris@myris.cz.
Subscribers of Springer need to access the articles on their site, which is https://link.springer.com/journal/10587.

[Previous Article] [Next Article] [Contents of This Number] [Contents of Czechoslovak Mathematical Journal]