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Journal of Function Spaces
Volume 2017, Article ID 2943073, 10 pages
https://doi.org/10.1155/2017/2943073
Research Article

The Characteristic Properties of the Minimal -Mean Width

College of Mathematics and Statistics, Hexi University, Zhangye, Gansu 734000, China

Correspondence should be addressed to Tongyi Ma; moc.621@iygnotam

Received 18 February 2017; Accepted 23 March 2017; Published 20 June 2017

Academic Editor: Antonio S. Granero

Copyright © 2017 Tongyi Ma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. A. A. Giannopoulos and V. D. Milman, “Extremal problems and isotropic positions of convex bodies,” Israel Journal of Mathematics, vol. 117, pp. 29–60, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. J. Yuan, G. Leng, and W.-S. Cheung, “Convex bodies with minimal p -mean width,” Houston Journal of Mathematics, vol. 36, no. 2, pp. 499–511, 2010. View at Google Scholar · View at MathSciNet · View at Scopus
  3. B. Fleury, O. Gu\'edon, and G. Paouris, “A stability result for mean width of Lp -centroid bodies,” Advances in Mathematics, vol. 214, no. 2, pp. 865–877, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. B. He, G. Leng, and K. Li, “Projection problems for symmetric polytopes,” Advances in Mathematics, vol. 207, no. 1, pp. 73–90, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. M. Ludwig, “Ellipsoids and matrix-valued valuations,” Duke Mathematical Journal, vol. 119, no. 1, pp. 159–188, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. M. Ludwig, “General affine surface areas,” Advances in Mathematics, vol. 224, no. 6, pp. 2346–2360, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. M. Ludwig and M. Reitzner, “A classification of SL(n) invariant valuations,” Annals of Mathematics, vol. 172, pp. 1219–1267, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  8. E. Lutwak, “The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem,” Journal of Differential Geometry, vol. 38, no. 1, pp. 131–150, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  9. E. Lutwak, “The Brunn-Minkowski-Firey theory. {II}. Affine and geominimal surface areas,” Advances in Mathematics, vol. 118, no. 2, pp. 244–294, 1996. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. E. Lutwak, D. Yang, and G. Zhang, “A new ellipsoid associated with convex bodies,” Duke Mathematical Journal, vol. 104, no. 3, pp. 375–390, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. E. Lutwak, D. Yang, and G. Zhang, “Lp affine isoperimetric inequalities,” Journal of Differential Geometry, vol. 56, no. 1, pp. 111–132, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. E. Lutwak, D. Yang, and G. Zhang, “A new affine invariant for polytopes and Schneider's projection problem,” Transactions of the American Mathematical Society, vol. 353, no. 5, pp. 1767–1779, 2001. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. E. Lutwak, D. Yang, and G. Zhang, “On the Lp -Minkowski problem,” Transactions of the American Mathematical Society, vol. 356, no. 11, pp. 4359–4370, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. E. Lutwak, D. Yang, and G. Zhang, “Lp John ellipsoids,” Proceedings of the London Mathematical Society, vol. 90, no. 2, pp. 497–520, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  15. T. Ma and Y. Feng, “The ith p-affine surface area,” Journal of Inequalities and Applications, vol. 2015, article 187, 26 pages. View at Publisher · View at Google Scholar · View at MathSciNet
  16. T. Ma, “The generalized Lp -Winternitz problem,” Journal of Mathematical Inequalities, vol. 9, no. 2, pp. 597–614, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. G. Paouris and E. M. Werner, “Relative entropy of cone measures and Lp centroid bodies,” Proceedings of the London Mathematical Society, vol. 104, no. 2, pp. 253–286, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  18. D. Ryabogin and A. Zvavitch, “The Fourier transform and FIRey projections of convex bodies,” Indiana University Mathematics Journal, vol. 53, no. 3, pp. 667–682, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. C. Schutt and E. Werner, “Surface bodies and p-affine surface area,” Advances in Mathematics, vol. 187, pp. 98–145, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  20. A. Stancu, “The discrete planar L0 -Minkowski problem,” Advances in Mathematics, vol. 167, no. 1, pp. 160–174, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. A. Stancu, “Centro-affine invariants for smooth convex bodies,” International Mathematics Research Notices, vol. 2012, no. 10, pp. 2289–2320, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  22. E. M. Werner, “R\'enyi divergence and Lp -affine surface area for convex bodies,” Advances in Mathematics, vol. 230, no. 3, pp. 1040–1059, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. E. Werner and D. Ye, “New Lp affine isoperimetric inequalities,” Advances in Mathematics, vol. 218, no. 3, pp. 762–780, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. G. Xiong, “Extremum problems for the cone volume functional of convex polytopes,” Advances in Mathematics, vol. 225, no. 6, pp. 3214–3228, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. V. Yaskin and M. Yaskina, “Centroid bodies and comparison of volumes,” Indiana University Mathematics Journal, vol. 55, no. 3, pp. 1175–1194, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. D. Zou and G. Xiong, “Orlicz-Legendre ellipsoids,” The Journal of Geometric Analysis, vol. 26, no. 3, pp. 2474–2502, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  27. J. Bastero and M. Romance, “Positions of convex bodies associated to extremal problems and isotropic measures,” Advances in Mathematics, vol. 184, no. 1, pp. 64–88, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  28. A. A. Giannopoulos and M. Papadimitrakis, “Isotropic surface area measures,” Mathematika, vol. 46, pp. 1–13, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  29. V. D. Milman and 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, vol. 1376 of Lecture Notes in Math, pp. 64–104, Springer, Berlin, Germany, 1989. View at Publisher · View at Google Scholar · View at MathSciNet
  30. G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge University Press, London, UK, 1934.
  31. B. Zhu, J. Zhou, and W. Xu, “Dual Orlicz-Brunn-Minkowski theory,” Advances in Mathematics, vol. 264, pp. 700–725, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  32. D. Zou and G. Xiong, “The minimal Orlicz surface area,” Advances in Applied Mathematics, vol. 61, pp. 25–45, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  33. A. J. Li and G. S. Leng, “Extremal problems related to Gauss-John position,” Acta Mathematica Sinica (English Series), vol. 28, no. 12, pp. 2527–2534, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus