Annals of Global Analysis and Geometry

, Volume 14, Issue 4, pp 373–380 | Cite as

A geometric inequality on mixed volumes

  • Young Do Chai


We develop some geometric inequality for a kind of generalized convex set. The integral of (n − 2)-th mean curvature of the generalized convex set, the mixed volume of the convex hull of the set, and a reference convex set are involved in the inequality.

Key words

Hyperplane section quasi-convex mixed volume an integral of mean curvature Minkowski functional 

MSC 1991

52 A 30 52 A 22 53 C 20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Bonnensen, T.; Fenchel, W.: Theorie der konvexen Körper. Springer, Berlin 1934.Google Scholar
  2. [2]
    Busemann, H.: Convex Surfaces. Interscience Publishers, Inc., New York 1958.Google Scholar
  3. [3]
    Chai, Y.D.: A topological charaterization of compact n-manifolds with C 2-boundary in R n. Topology Appl. 43 (1992), 83–89.Google Scholar
  4. [4]
    Chern, S.S.: On the kinematic formula in the Euclidean space of n-dimensions. Am. J. Math. 74 (1952), 227–236.Google Scholar
  5. [5]
    Firey, W.J.: The mixed area of a convex body and its polar reciprocal. Isr. J. Math. 1 (1963), 201–202.Google Scholar
  6. [6]
    Flanders, H.: A proof of Minkowski's inequality for convex curves. Am. Math. Mon. 75 (1968), 581–593.Google Scholar
  7. [7]
    Ghandehari, M.: Polar duals of convex bodies. Proc. Am. Math. Soc. 113 (1991), 799–808.Google Scholar
  8. [8]
    Minkowski, H.: Theorie der konvexen Körper, insbesondere Begründung ihres Oberflächenbegriffs. Ges. Abhandl., Leipzig-Berlin 2 (1911), 131–229.Google Scholar
  9. [9]
    Osserman, R.: Bonnesen-style isoperimetric inequalities. Am. Math. Mon. 86 (1979), 1–29.Google Scholar
  10. [10]
    Polya, G.; Szegö, G.: Isoperimetric inequalities in mathematical physics. Ann. Math. Studies 27, 1951.Google Scholar
  11. [11]
    Santalo, L.A.: Integral Geometry and Geometric Probability. Addison-Wesley Publishing Co., 1976.Google Scholar
  12. [12]
    Schneider, R.: Convex bodies: The Brunn-Minkowski theory. Cambridge University Press, 1993.Google Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Young Do Chai
    • 1
  1. 1.Department of MathematicsSung Kyun Kwan UniversitySuwonSouth Korea

Personalised recommendations