The Brunn-Minkowski Inequalities for Centroid Body

DOI: 10.4236/apm.2013.31013   PDF   HTML   XML   3,703 Downloads   5,474 Views  


In [1], the authors established the Brunn-Minkowski inequality for centroid body. In this paper, we give an isolate form and volume difference of it, respectively. Both of these results are strength versions of the original.

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J. Yuan and L. Zhao, "The Brunn-Minkowski Inequalities for Centroid Body," Advances in Pure Mathematics, Vol. 3 No. 1, 2013, pp. 105-108. doi: 10.4236/apm.2013.31013.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] J. Yuan, L. Z. Zhao and G. S. Leng, “Inequalities for Lp Centroid Body,” Taiwanese Journal of Mathematics, Vol. 11, No. 5, 2007, pp. 1315-1325.
[2] C. M. Petty, “Centroid Surface,” Pacific Journal of Mathematics, Vol. 11, No. 4, 1961, pp. 1535-1547. doi:10.2140/pjm.1961.11.1535
[3] K. Leichtwei, “Affine Geometry of Convex Bodies,” J. A. Barth, Heidelberg, 1998.
[4] R. Schneider, “Convex Bodies: The Brunn-Minkowski Theory,” Cambridge University Press, Cambridge, 1993. doi:10.1017/CBO9780511526282
[5] R. J. Gardner, “Geometric Tomography,” Cambridge University Press, Cambridge, 1995.
[6] E. Lutwak, “Centroid Bodies and Dual Mixed Volumes,” Proceedings London Mathematical Society, Vol. 60, No. 2, 1990, pp. 365-391. doi:10.1112/plms/s3-60.2.365
[7] G. Y. Zhang, “Centered Bodies and Dual Mixed Volumes,” Transactions of the American Mathematical Society, Vol. 345, No. 2, 1994, pp. 777-801. doi:10.1090/S0002-9947-1994-1254193-9
[8] E. Lutwak, “The Brunn-Minkowski-Firey Theory I: Mixed Volumes and the Minkowski Problem,” Journal of Differential Geometry, Vol. 38, 1993, pp. 131-150.
[9] A. D. Aleksandrov, “Zur Theorie der Gemischten Volumina von Konvexen K?rpern, I. Verallgemeinerung einiger Begriffe der Theorie der Konvexen Korpern,” Matematicheskii Sbornik, Vol. 2, 1937, pp. 947-972.
[10] W. Fenchel and B. Jessen, “Mengenfunktionen und Konvexe K?rpern,” Danske Videnskabernes Selskab. Matematisk-Fysiske Meddelelser, Vol. 16, 1938, p. 3.
[11] R. J. Gardner, “The Brunn-Minkowski Inequality,” Bulletin of the American Mathematical Society, Vol. 39, No. 3, 2002, pp. 355-405. doi:10.1090/S0273-0979-02-00941-2
[12] E. F. Beckenbach and R. Bellman, “Inequalities,” Springer, Berlin, 1961. doi:10.1007/978-3-642-64971-4

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