Publication Cover
Applicable Analysis
An International Journal
Volume 101, 2022 - Issue 9
87
Views
0
CrossRef citations to date
0
Altmetric
Research Article

The minimal Orlicz mean width of convex bodies

Pages 3316-3346 | Received 24 Jan 2019, Accepted 19 Jun 2020, Published online: 17 Nov 2020

References

  • Giannopoulos AA, Milman VD. Extremal problems and isotropic positions of convex bodies. Israel J Math. 2000;117:29–60.
  • Yuan J, Leng G, Cheung W. Convex bodies with minimal p-mean width. Houston J Math. 2010;36(2):499–511.
  • Ma T. The characteristic properties of the minimal Lp-mean width. J Function Spaces. 2017;2017. Article ID 2943073, 10 pages. Available at https://www.hindawi.com/journals/jfs/2017/2943073/.
  • Feng Y, Wang W. Blaschke-Minkowski homomorphisms and affine surface area. Publ Math Debrecen. 2014;85(3-4):297–308.
  • Fleury B, Guédon O, Paouris GA. A stability result for mean width of Lp-centroid bodies. Adv Math. 2007;214:865–877.
  • He B, Leng G, Li K. Projection problems for symmetric polytopes. Adv Math. 2006;207:73–90.
  • Huang Y, Lutwak E, Yang D, et al. Geometric measures in the dual Brunn-Minkowski theory and their associated Minkowski problems. Acta Math. 2016;216:325–388.
  • Ludwig M. Ellipsoids and matrix-valued valuations. Duke Math J. 2003;119:159–188.
  • Ludwig M. General affine surface areas. Adv Math. 2010;224:2346–2360.
  • Ludwig M, Reitzner M. A classification of SL(n) invariant valuations. Ann Math. 2010;172:1219–1267.
  • Lutwak E. The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem. J Differential Geom. 1993;38:131–150.
  • Lutwak E. The Brunn-Minkowski-Firey theory. II. Affine and geominimal surface areas. Adv Math. 1996;118:244–294.
  • Lutwak E, Yang D, Zhang G. A new ellipsoid associated with convex bodies. Duke Math. J. 2000;104:375–390.
  • Lutwak E, Yang D, Zhang G. Lp affine isoperimetric inequalities. J Differential Geom. 2000;56:111–132.
  • Lutwak E, Yang D, Zhang G. A new affine invariant for polytopes and Schneider's projection problem. Trans Amer Math Soc. 2001;353:1767–1779.
  • Lutwak E, Yang D, Zhang G. On the Lp-Minkowski problem. Trans Amer Math Soc. 2004;356:4359–4370.
  • Lutwak E, Yang D, Zhang G. Lp John ellipsoids. Proc Lond Math Soc. 2005;90:497–520.
  • Lutwak E, Yang D, Zhang G. Lp dual curvature measures. Adv Math. 2018;329:85–132.
  • Ma T, Feng Y. The ith p-affine surface area. J Ineq Appl. 2015;2015(187):1–26.
  • Ma T. The generalized Lp-Winternitz problem. J Math Inequal. 2015;9(2):597–614.
  • Paouris G, Werner E. Relative entropy of cone-volumes and Lp centroid bodies. Proc Lond Math Soc. 2012;104:253–286.
  • Ryabogin D, Zvavitch A. The Fourier transform and firey projections of convex bodies. Indiana Univ Math J. 2004;53:667–682.
  • Schütt C, Werner E. Surface bodies and p-affine surface area. Adv Math. 2004;187:98–145.
  • Stancu A. The discrete planar L0-Minkowski problem. Adv Math. 2002;167:160–174.
  • Stancu A. Centro-affine invariants for smooth convex bodies. Int Math Res Not. 2012;180:2289–2320.
  • Werner E. Rényi divergence and Lp-affine surface area for convex bodies. Adv Math. 2012;230:1040–1059.
  • Werner E, Ye D. New Lp affine isoperimetric inequalities. Adv Math. 2008;218:762–780.
  • Wu D. A generalization of Lp-Brunn-Minkowski inequalities and Lp-Minkowski problems for measures. Adv Appl Math. 2017;89:156–183.
  • Xiong G. Extremum problems for the cone volume functional of convex polytopes. Adv Math. 2010;225:3214–3228.
  • Yaskin V, Yaskina M. Centroid bodies and comparison of volumes. Indiana Univ Math J. 2006;55:1175–1194.
  • Gardner RJ, Hu D, Weil W. The Orlicz-Brunn-Minkowski theory: a general framework, additions, and inequalities. J Differential Geom. 2014;97:427–476.
  • Haberl C, Lutwak E, Yang D, et al. The even Orlicz Minkowski problem. Adv Math. 2010;224:2485–2510.
  • Li A, Leng G. A new proof of the Orlicz Busemann-Petty centroid inequality. Proc Amer Math Soc. 2011;139:1473–1481.
  • Lin Y. Affine Orlicz Pólya-Szegö principle for log-concave functions. J Funct Anal. 2017;273:3295–3326.
  • Lutwak E, Yang D, Zhang G. Orlicz projection bodies. Adv Math. 2010;223:220–242.
  • Lutwak E, Yang D, Zhang G. Orlicz centroid bodies. J Differential Geom. 2010;84:365–387.
  • Ma T, Wang W. Dual Orlicz geominimal surface area. J Ineq Appl. 2016;2016(56):1–13.
  • Ma T. The minimal dual Orlicz surface area. Taiwan J Math. 2016;20(2):287–309.
  • Ma T. On the reverse Orlicz Blaschke-Santaló inequality. Mediterr J Math. 2018;15(32):1–11.
  • Xi D, Jin H, Leng G. The Orlicz Brunn-Minkowski inequality. Adv Math. 2014;260:350–374.
  • Xiong G, Zou D. Orlicz mixed quermassintegrals. Sci China Math. 2014;57:1–14.
  • Zhu G. The Orlicz centroid inequality for star bodies. Adv Appl Math. 2012;48:432–445.
  • Zhu B, Zhou J, Xu W. Dual Orlicz-Brunn-Minkowski theory. Adv Math. 2014;264:700–725.
  • Zou D, Xiong G. Orlicz-John ellipsoids. Adv Math. 2014;265:132–168.
  • Zou D, Xiong G. Orlicz-Legendre ellipsoids. J Geom Anal. 2014;26:1–29.
  • Zou D, Xiong G. The minimal Orlicz surface area. Adv Appl Math. 2014;61:25–45.
  • Gruber PM, Schuster F. An arithmetic proof of John's ellipsoid theorem. Arch Math. 2005;85:82–88.
  • Gruber PM. John and Loewner ellipsoids. Discrete Comput Geom. 2011;46:776–788.
  • Firey W. Polar means of convex bodies and a dual to the Brunn-Minkowski theorem. Canad J Math. 1961;13:444–453.
  • Firey W. Mean cross-section measures of harmonic means of convex bodies. Pacific J Math. 1961;11:1263–1266.
  • Lutwak E, Zhang G. Blaschke-Santaló's inequalities. J Differential Geom. 1997;47:1–16.
  • Ma T, Zhang D. Lp-centroid bodies and its characterizations. Commun Math Res. 2015;31(11):1775–1788.
  • Bastero J, Romance M. Positions of convex bodies associated to extremal problems and isotropic measures. Adv Math. 2004;184:64–88.
  • Giannopoulos AA, Papadimitrakis M. Isotropic surface area measures. Mathematika. 1999;46:1–13.
  • Milman VD, Pajor A. Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space. In: Geometric aspects of functional analysis. Berlin: Springer; 1989. p. 64–104. (Lecture Notes in Mathematics; 1376).
  • Groemer H. Geometric applications of fourier series and spherical harmonics. Cambridge: Cambridge University Press; 1996. (Encyclopedia of Mathematics and its Applications; 61).
  • Hardy GH, Littlewood JE, Pólya G. Inequalities. London: Cambridge University Press; 1934.
  • Giannopoulos AA, Milman VD, Rudelson M. Convex bodies with minimal mean width. In: Geometric aspects of functional analysis. Berlin: Springer; 2000. p. 81–93. (Lecture Notes in Mathematics; 1745).
  • Wang S, Wu M, Jia Z. Matrix inequality. 2nd ed. Beijing: Science Press; 1992. (in Chinese).
  • Xia D, Wu Z, Yan S, et al. Theory of function of real variable and functional analysis. 2nd ed. Beijing: Higher Education Press in China; 2010. (in Chinese).
  • Wu S. Generalization of Hölder inequality and Minkowski inequality. Acta Math Sinca, Chinese Ser. 2006;49(6):1267–1274. (in Chinese).
  • Schneider R. Convex bodies: the Brunn-Minkowski theory. 2nd ed. New York: Cambridge University Press; 2014.
  • Gel'fand IM. Lectures on linear algebra. New York: Interscience; 1967.
  • Feng D. The basis of convex analysis. Beijing: Science Press; 1995. (in Chinese).
  • Hu L, Men Z. Convex analysis and non-smooth analysis. Shanghai: Shanghai Science and Technology Press; 2000. (in Chinese).
  • Gruber PM. Convex and discrete geometry. Berlin: Springer; 2007. (Grundlehren der Mathematischen Wissenschaften; 336).
  • Gruber PM. Application of an idea of Voronoi to John type problems. Adv Math. 2008;218:309–351.
  • Stefanakis I. A note on the surface isotropic position. File available at www.math.uoc.gr/papadim/Isotropic.ps.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.