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MIXED BRIGHTNESS-INTEGRALS OF CONVEX BODIES

  • Li, Ni (COLLEGE OF MATHEMATICS AND COMPUTER SCIENCE CHONGQING NORMAL UNIVERSITY) ;
  • Zhu, Baocheng (COLLEGE OF MATHEMATICS AND COMPUTER SCIENCE CHONGQING NORMAL UNIVERSITY)
  • Received : 2008.12.08
  • Published : 2010.09.01

Abstract

The mixed width-integrals of convex bodies are defined by E. Lutwak. In this paper, the mixed brightness-integrals of convex bodies are defined. An inequality is established for the mixed brightness-integrals analogous to the Fenchel-Aleksandrov inequality for the mixed volumes. An isoperimetric inequality (involving the mixed brightness-integrals) is presented which generalizes an inequality recently obtained by Chakerian and Heil. Strengthened version of this general inequality is obtained by introducing indexed mixed brightness-integrals.

Keywords

convex bodies;mixed projection bodies;brightness;mixed brightness;mixed brightness-integrals;Fenchel-Aleksandrov inequality

Acknowledgement

Supported by : Nature Science Foundation of China

References

  1. G. D. Chakerian, Isoperimetric inequalities for the mean width of a convex body, Geometriae Dedicata 1 (1973), no. 3, 356-362.
  2. G. D. Chakerian, The mean volume of boxes and cylinders circumscribed about a convex body, Israel J. Math. 12 (1972), 249-256. https://doi.org/10.1007/BF02790751
  3. R. J. Gardner, Geometric Tomography, Encyclopedia of Mathematics and its Applications, 58. Cambridge University Press, Cambridge, 1995.
  4. H. Hardy, J. E. Littlewood, and G. P´olya, Inequalities, Reprint of the 1952 edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1988.
  5. F. Lu, Mixed chord-integral of star bodies, preprint, 2007.
  6. E. Lutwak, Width-integrals of convex bodies, Proc. Amer. Math. Soc. 53 (1975), no. 2, 435-439. https://doi.org/10.1090/S0002-9939-1975-0383254-5
  7. E. Lutwak, A general Bieberbach inequality, Math. Proc. Cambridge Philos. Soc. 78 (1975), no. 3, 493-495. https://doi.org/10.1017/S0305004100051963
  8. E. Lutwak, Mixed width-integrals of convex bodies, Israel J. Math. 28 (1977), no. 3, 249-253. https://doi.org/10.1007/BF02759811
  9. E. Lutwak, Mixed projection inequalities, Trans. Amer. Math. Soc. 287 (1985), no. 1, 91-105. https://doi.org/10.1090/S0002-9947-1985-0766208-7
  10. E. Lutwak, Inequalities for mixed projection bodies, Trans. Amer. Math. Soc. 339 (1993), no. 2, 901-916. https://doi.org/10.2307/2154305
  11. R. Schneider, Convex Bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and its Applications, 44. Cambridge University Press, Cambridge, 1993.

Cited by

  1. Inequalities for mixed width-integrals vol.21, pp.3, 2016, https://doi.org/10.1007/s11859-016-1157-6
  2. The Brunn-Minkowski type inequalities for mixed brightness-integrals vol.19, pp.4, 2014, https://doi.org/10.1007/s11859-014-1013-5
  3. General L p $L_{p}$ -mixed-brightness integrals vol.2015, pp.1, 2015, https://doi.org/10.1186/s13660-015-0708-2