DOI QR코드

DOI QR Code

A GEOMETRIC INEQUALITY ON A COMPACT DOMAIN IN ℝn

  • Chai, Young Do (Department of Mathematics College of Natural Sciences Sungkyunkwan University) ;
  • Cho, Yong Seung (Division of Mathematical and Physical Science College of Natural Sciences Ewha Womans University)
  • Received : 2016.02.01
  • Accepted : 2017.08.10
  • Published : 2018.01.31

Abstract

In this paper, we study some topological structure of a compact domain in ${\mathbb{R}}^n$ in terms of the curvature conditions and develop a geometric inequality involving the volume and the integral of mean curvatures over the boundary of the compact domain.

Acknowledgement

Supported by : National Research Foundation of Korea (NRF)

References

  1. Y. D. Chai, A new lower bound for the integral of the (n-2)nd mean curvature over the boundary of a compact domain in $-\mathbb-R}}^n$, Differential Geom. Appl. 7 (1997), no. 1, 35-40. https://doi.org/10.1016/S0926-2245(96)00028-9
  2. Y. D. Chai and G. Kim, A Characterization of compact sets in $-\mathbb-R}}^n$ and its Application to a geometric inequality, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 17 (2010), no. 4, 543-553.
  3. S. S. Chern, Global differential geometry, MAA 27 (1989), 303-350.
  4. H. Flanders, A proof of Minkowski's inequality for convex curves, Amer. Math. Monthly 75 (1968), 581-593. https://doi.org/10.1080/00029890.1968.11971034
  5. M. Gromov, Hyperbolic manifolds, group and actions, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), pp. 183-213, Ann. of Math. Stud., 97, Princeton Univ. Press, Princeton, N.J., 1981.
  6. J. Milnor, Morse Theory, Annals of Mathematics Studies, No. 51 Princeton University Press, Princeton, N. J., 1963.
  7. Polya and Szego, Isoperimetric inequalities in mathematical physics, Annals of Mathematics Studies, no. 27, Princeton University Press, Princeton, N. J., 1951.
  8. A. Ros, Compact hypersurfaces with constant higher order mean curvatures, Rev. Mat. Iberoamericana 3 (1987), no. 3-4, 447-453.
  9. L. A. Santalo, Integral Geometry and Geometric Probability, Addison-Wesley Publishing Co., 1976.
  10. J. P. Sha, p-convex Riemannian manifolds, Invent. Math. 83 (1986), no. 3, 437-447. https://doi.org/10.1007/BF01394417
  11. F. A. Valentine, Convex Sets, Mcgraw-Hill Book Co., 1964.