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An Extremal Property of p-mean Width

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Abstract

Several isoperimetric type inequalities for p-mean width of convex bodies in \(\mathbb {R}^n\) are established. These inequalities show the interrelations among the p-mean width of a convex body in \(\mathbb {R}^n\), an isotropic measure on unit sphere, and the newly-introduced \(L_{r,s}\)-pseudo-moment body of the given body in \(\mathbb {R}^n\). The equalities in these inequalities are all characterized by parallelotopes.

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References

  1. Ball, K.: Volumes of sections of cubes and related problems, Geometric aspects of functional analysis (J. Lindenstrauss and V.D. Milman, Eds.), Springer Lecture Notes in Math. 1376, pp. 251–260 (1989)

  2. Ball, K.: Volume ratios and a reverse isoperimetric inequality. J. Lond. Math. Soc. 44(2), 351–359 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ball, K.: Shadows of convex bodies. Trans. Am. Math. Soc. 327, 891–901 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  4. Barthe, F.: On a reverse form of the Brascamp–Lieb inequality. Invent. Math. 134, 335–361 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. Barthe, F.: An extremal property of the mean width of the simplex. Math. Ann. 310, 685–693 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. Gardner, R.J.: Geometric Tomography, Encyclopedia of Mathematics and its Applications, vol. 58. Cambridge University Press, New York (2006)

    Google Scholar 

  7. Giannopoulos, A., Papadimitrakis, M.: Isotropic surface area measures. Mathematika 46, 1–13 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gruber, P.M.: Convex and Discrete Geometry, Grundlehren der Mathematischen Wisenschaften, vol. 336. Springer, Berlin-Heidelberg (2007)

    Google Scholar 

  9. Huang, Q., He, B.: Gaussian inequalities for Wulff shapes. Geom. Dedic. 169, 33–47 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Li, A.J., Leng, G.: Mean width inequalities for isotropic measures. Math. Z. 270, 1089–1110 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Li, A.J., Huang, Q.: The \(L_p\) Loomis–Whitney inequality. Adv. Appl. Math. 75, 94–115 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Li, A.J., Huang, Q.: The dual Loomis–Whitney inequality. Bull. Lond. Math. Soc. 48, 676–690 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lieb, E.L., Loss, M.: Analysis, 2nd edition, Graduate studies in Mathematics, vol. 14. American Mathematical Society, Providence (2001)

    Google Scholar 

  14. Loomis, L.H., Whitney, H.: An inequality related to the isoperimetric inequality. Bull. Am. Math. Soc. 55, 961–962 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lutwak, E.: The Brunn–Minkowski–Firey theory. I. Mixed volumes and the Minkowski problem. J. Differ. Geom. 38, 131–150 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lutwak, E.: The Brunn–Minkowski–Firey theory. II. Affine and geominimal surface areas. Adv. Math. 118, 244–294 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lutwak, E., Yang, D., Zhang, G.: \(L_p\) affine isoperimetric inequality. J. Differ. Geom. 56, 111–132 (2000)

    Article  MATH  Google Scholar 

  18. Lutwak, E., Yang, D., Zhang, G.: Volume inequalities for subspaces of \(L_p\). J. Differ. Geom. 68, 159–184 (2004)

    Article  MATH  Google Scholar 

  19. Lutwak, E., Yang, D., Zhang, G.: Volume inequalities for isotropic measures. Am. Math. J. 129, 1711–1723 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lutwak, E., Yang, D., Zhang, G.: A volume inequality for polar bodies. J. Differ. Geom. 84(1), 163–178 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lutwak, E., Zhang, G.: Blaschke–Santalo inequalities. J. Differ. Geom. 47, 1–16 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  22. Meyer, M.: A volume inequality concerning sections of convex sets. Bull. Lond. Math. Soc. 20, 151–155 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  23. Meyer, M., Pajor, A.: On the Blaschke–Santal inequality. Arch. Math. (Basel) 55, 82–93 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  24. Schuster, F.E., Weberndorfer, M.: Volume inequalities for asymmetric Wulff shapes. J. Differ. Geom. 92, 263–283 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Schneider, R.: Convex bodies: The Brunn–Minkowski theory, 2nd Edn, Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2014)

    Google Scholar 

  26. Schechtman, G., Schmuckenschläger, M.: (1995) A concentration inequality for harmonic measures on the sphere, GAFA Seminar ed. by J. Lindenstrauss and V. Milman, Operator Theory Advances and Applications 77, 255-274

  27. Schmuckenschläger, M.: (1999) An extremal property of the regular simplex. In: Convex Geometric Analysis (Berkeley, CA, 1996), pp. 199–202. Math. Sci. Res. Inst. Publ., vol. 34 Cambridge University Press, Cambridge

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Authors and Affiliations

  1. School of Mathematical Sciences, Chongqing Normal University, Chongqing, 401331, China

    Qunli Long & Songjun Lv

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  1. Qunli Long
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  2. Songjun Lv
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Correspondence to Songjun Lv.

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Research supported partly by NSFC under Grant 10801140, CSTC under Grant 2013-JCYJ-A00005, CQNU Foundation under Grant 13XLZ05.

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Long, Q., Lv, S. An Extremal Property of p-mean Width. Results Math 73, 26 (2018). https://doi.org/10.1007/s00025-018-0786-9

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