Abstract
Recently, the connection between p-measures of asymmetry and the Lp -mixed volumes for convex bodies was found soon after the p-measure of asymmetry was proposed, and the Orlicz-measures of asymmetry was proposed inspired by such a kind of connection. In this paper, by a similar way the dual p -measures of asymmetry for star bodies (naturally for convex bodies) is introduced first. Then the connection between dual p -measures of asymmetry and Lp -dual mixed volumes is established. Finally, the best lower and upper bounds of dual p-measures and the corresponding extremal bodies are discussed.
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References
Minkowski H. Allegemeine Lehrsätze über die convexen polyeder[J]. Nachr Ges Wiss Göttingen, 1897,(2): 198–219.
Grünbaum B. Measures of symmetry for convex sets [C]//Proc Symp Pure Math, Vol., Providence: Amer Math Soc, 1963: 233–270.
Guo Q. Stability of the Minkowski measure of asymmetry for convex bodies[J]. Discrete Comput Geom, 2005, 34: 351–362.
Guo Q, Kaijser S. On asymmetry of some convex bodies[J]. Discrete Comput Geom, 2002, 27: 239–247.
Jin H, Guo Q. On the asymmetry for convex domains of constant width[J]. Commun Math Res, 2010, 26(2):176–182.
Jin H L, Guo Q. Asymmetry of convex bodies of constant width[J]. Discrete Comput Geom, 2012, 47: 415–423.
Jin H L, Guo Q. A note on the extremal bodies of constant width for the Minkowski measure [J]. Geom Dedicata, 2013, 164: 227–229.
Guo Q, Guo J F, Su X L. The measures of asymmetry for coproducts of convex bodies [J]. Pacific J Math, 2015, 276: 401–418.
Toth G. Measures of Symmetry for Convex Sets and Stability[M]. Berlin: Springer–Verlag, 2015.
Guo Q. On p–measures of asymmetry for convex bodies[J]. Adv Geom, 2012, 12(2): 287–301.
Guo Q, Toth G. Dual mean Minkowski measures of symmetry for convex bodies [J]. Science China: Mathematics, 2016, 59(7): 1383–1394.
Guo Q, Toth G. Dual mean Minkowski measures and the Grünbaum conjecture for affine diameters[J]. Pacific J Math, 2017, 292: 117–137.
Jin H L, Leng G S, Guo Q. Mixed volumes and measures of asymmetry [J]. Acta Math Sinica (Eng Series), 2014, 30(11): 1905–1916.
Jin H L. The log–Minkowski measures of asymmetry for convex bodies [J/OL]. Geom Dedicata, http://doi.org.10./1007 /s10711–017–0302–5, 2017.
Meyer M, Schütt C, Werner E M. New affine measures of symmetry for convex bodies [J]. Adv Math, 2011, 228: 2920–2942.
Guédon O, Litvak A E. On the symmetric average of a convex body [J]. Adv Geom, 2011, 11: 615–622.
Toth G. A measure of symmetry for the moduli of spherical minimal immersions [J]. Geom Dedicata, 2012, 160: 1–14.
Toth G. On the shape of the moduli of spherical minimal immersions[J]. Trans Amer Math Soc, 2006, 358(6): 2425–2446.
Toth G. Asymmetry of convex sets with isolated extreme points[J]. Proc Amer Math Soc, 2009, 137: 287–295.
Schneider R. Convex Bodies: The Brunn–Minkowski Theory [M]. 2nd ed. Cambridge: Camb Univ Press, 2014.
Lutwak E. The Brunn–Minkowski–Firey theory. II. Affine and geominimal surface areas[J]. Adv Math, 1996, 118(2): 244–294.
Lutwak E, Yang D, Zhang G. p L John ellipsoids[J]. Proc London Math Soc, 2005, 90(2): 497–520.
Shao Y C, Guo Q. The equivalence and estimates of several affine invariant distances between convex bodies [J]. J of Math (PRC), 2015, 35(2): 287–293(Ch).
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Foundation item: Supported by the National Natural Science Foundation of China (12671293, 11701118, U1201252), the National High Technology Research & Development Program of China (2015AA015408), the Special Fund for Science & Technology Platform and Talent Team Project of Guizhou Province (QianKe-HePingTai RenCai [2016]5609)
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Huang, X., Zhu, H. & Guo, Q. On the Dual p-Measures of Asymmetry for Star Bodies. Wuhan Univ. J. Nat. Sci. 23, 465–470 (2018). https://doi.org/10.1007/s11859-018-1349-3
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DOI: https://doi.org/10.1007/s11859-018-1349-3