Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

INEQUALITIES FOR DUAL HARMONIC QUERMASSINTEGRALS

  • Jun, Yuan (Department of Mathematics Shanghai University) ;
  • Shufeng Yuan (Department of Mathematics Shanghai University) ;
  • Gangsong Leng (Department of Mathematics Shanghai University)
  • Published : 2006.05.01
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we study the properties of the dual harmonic quermassintegrals systematically and establish some inequalities for the dual harmonic quermassintegrals, such as the Minkowski inequality, the Brunn-Minkowski inequality, the Blaschke-Santalo inequality and the Bieberbach inequality.

Keywords

References

  1. W. Blaschke, Uber affine Geometrie IX: Verschiedene Bemerkungen und Auf- gaben. Ber. Verh. Sachs. Akad. Wiss. Leipzig Math.-Phys. Kl 69 (1917), 412-420
  2. L. Bieberbach, Uber eine extremaleigenschaft des kreises. Jber. Deutsch. Math.-Verein. 24 (1915), 247-250
  3. T. Bonnesen and W. Fenchel, Theorie der konvexen Korper, Springer, Berlin, 1934
  4. R. J. Gardner, Geometric tomography, Cambridge University Press, New York, 1995
  5. H. Hadwiger and H. Sagan, Vorlesungen uber Inhalt, Oberflache und Isoperime- trie, Amer. Math. Monthly 65 (1958), no. 4, 300
  6. D. Klain, Star valuations and dual mixed volumes, Adv. Math. 121 (1996), no. 1, 80-101 https://doi.org/10.1006/aima.1996.0048
  7. K. Leichtweiss. Konvexe Mengen, Springer-Verlag, Berlin-New York, 1980
  8. E. Lutwak, Dual mixed volumes, Pacific J. Math. 58 (1975), no. 2, 531-538 https://doi.org/10.2140/pjm.1975.58.531
  9. E. Lutwak, A general Bieberbach inequality, Math. Proc. Cambrigde Philos. Soc. 78 (1975), 493-495
  10. E. Lutwak, A general isepiphanic inequality, Proc. Amer. Math. Soc. 90 (1984), no. 3, 415-421
  11. E. Lutwak, Mean dual and harmonic cross-sectional measures, Ann. Mat. Pura Appl. (4) 119 (1979), 139-148 https://doi.org/10.1007/BF02413172
  12. E. Lutwak, Inequalities for Hadwiger's harmonic quermassintegrals, Math. Ann. 280 (1988), no. 1, 165-175 https://doi.org/10.1007/BF01474188
  13. E. Lutwak, Intersection bodies and dual mixed volumes, Adv. Math. 71 (1988), no. 2, 232-261 https://doi.org/10.1016/0001-8708(88)90077-1
  14. M. Maria, Quotient star bodies, intersection bodies and star duality. J. Math. Anal. Appl. 232 (1999), no. 1, 45-60 https://doi.org/10.1006/jmaa.1998.6238
  15. L. A. Santalo, Un invariante afin para los cuerpos del espacio de n dimensiones. Portugal. Math. 8 (1949), 155-161
  16. L. A. Santalo, Integral geometry and geometric probability. Reading, MA: Addison- Wesley, 1976
  17. P. Urysohn, Mean width and volume of bodies in n dimensional space, Rec. Math. Soc. Math. Moscow 8 (1924), 477-480.(Russian)
  18. R. Schneider, Convex bodies: the Brunn-Minkowski theory, Cambridge University Press, Cambridge, 1993
  19. G. Y. Zhang, Dual kinematic formulas, Tran. Amer. Math. 351 (1999), no. 3, 985-995 https://doi.org/10.1090/S0002-9947-99-02053-X