대한수학회지 (Journal of the Korean Mathematical Society)

  • 제45권2호
  • /
  • Pages.587-596
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  • 2008
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  • 0304-9914(pISSN)
  • /
  • 2234-3008(eISSN)
대한수학회 (Korean Mathematical Society)
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INEQUALITIES FOR CHORD POWER INTEGRALS

  • Xiong, Ge (DEPARTMENT OF MATHEMATICS SHANGHAI UNIVERSITY) ;
  • Song, Xiaogang (DEPARTMENT OF MATHEMATICS SHANGHAI UNIVERSITY)
  • 발행 : 2008.03.31
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초록

For convex bodies, chord power integrals were introduced and studied in several papers (see [3], [6], [14], [15], etc.). The aim of this article is to study them further, that is, we establish the Brunn-Minkowski-type inequalities and get the upper bound for chord power integrals of convex bodies. Finally, we get the famous Zhang projection inequality as a corollary. Here, it is deserved to mention that we make use of a completely distinct method, that is using the theory of inclusion measure, to establish the inequality.

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참고문헌

  1. A. D. Alexandrov, On the theory of mixed volumes of convex bodies, Mat. Sb. 3 (1938), no. 45, 28-46
  2. Yu. D. Burago and V. A. Zalgaller, Geometric Inequalities, Springer, Berlin, 1980
  3. G. D. Chakerian, Inequalities for the difference body of a convex body, Proc. Amer. Math. Soc. 18 (1967), 879-884 https://doi.org/10.2307/2035131
  4. H. G. Eggleston, B. Grunbaum, and V. Klee, Some semicontinuity theorems for convex polytopes and cell-complexes, Comment. Math. Helv. 39 (1964), 165-188 https://doi.org/10.1007/BF02566949
  5. R. J. Gardner and G. Zhang, Affine inequalities and radial mean bodies, Amer. J. Math. 120 (1998), no. 3, 505-528 https://doi.org/10.1353/ajm.1998.0021
  6. D. Ren, An Introduction to Integral Geometry, Science and Technology Press, Shanghai, 1988
  7. L. A. Santalo, Integral Geometry and Geometric Probability, With a foreword by Mark Kac. Encyclopedia of Mathematics and its Applications, Vol. 1. Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976
  8. G. Xiong, Wing-Sum Cheung, and De-Yi Li, Bounds for inclusion measures of convex bodies, Adv. Appl. Math., in press, 2008 https://doi.org/10.1016/j.aam.2007.09.004
  9. G. Xiong, D. Li, and L. Zheng, Intersection bodies and inclusion measures of convex bodies, J. Math. (Wuhan) 23 (2003), no. 1, 6-10
  10. G. Xiong, J. Ni, and Z. Ruan, Inclusion measures of convex bodies, J. Math. Res. Exp. 27 (2007), no. 1, 81-86
  11. G. Zhang, Integral geometric inequalities, Acta Math. Sinica 34 (1991), no. 1, 72-90
  12. G. Zhang, Restricted chord projection and affine inequalities, Geom. Dedicata 39 (1991), no. 2, 213-222
  13. G. Zhang, Geometric inequalities and inclusion measures of convex bodies, Mathematika 41 (1994), no. 1, 95-116 https://doi.org/10.1112/S0025579300007208
  14. G. Zhang, Dual kinematic formulas, Trans. Amer. Math. Soc. 351 (1999), no. 3, 985-995 https://doi.org/10.1090/S0002-9947-99-02053-X
  15. G. Zhang and J. Zhou, Containment Measures in Integral Geometry, Integral geometry and convexity, 153-168, World Sci. Publ., Hackensack, NJ, 2006

피인용 문헌

  1. CHORD POWER INTEGRALS OF SIMPLICES vol.02, pp.04, 2009, https://doi.org/10.1142/S1793557109000479
  2. Random chord distributions and containment functions vol.58, 2014, https://doi.org/10.1016/j.aam.2014.05.003
  3. Some inequalities for chord power integrals of parallelotopes vol.181, pp.4, 2016, https://doi.org/10.1007/s00605-016-0888-y