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A geometric inequality on a compact domain in $\mathbb R^{n}$

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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.

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.

참고문헌 (11)

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  2. Y. D. Chai, "A Characterization of compact sets in <TEX>$-\mathbb-R}}^n$<", TEX> and its Application to a geometric inequality, 2010, pp.17(4), 543~553
  3. S. S. Chern, "Global differential geometry", MAA, 27, pp.303~350 (1989)
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  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)", Ann. of Math. Stud., 97, pp.183~213 (1981)
  6. J. Milnor, "Morse Theory", Annals of Mathematics Studies, (51) (1963)
  7. Polya, "Isoperimetric inequalities in mathematical physics", Annals of Mathematics Studies, (27) (1951)
  8. A. Ros, "Compact hypersurfaces with constant higher order mean curvatures", Rev. Mat. Iberoamericana, 3(3-4), pp.447~453 (1987)
  9. L. A. Santalo, "Integral Geometry and Geometric Probability" (1976), Addison-Wesley Publishing Co.
  10. J. P. Sha, "p-convex Riemannian manifolds", Invent. Math., 83(3), pp.437~447 (1986)[DOI]
  11. F. A. Valentine, "Convex Sets" (1964), Mcgraw-Hill Book Co.

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