Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
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COMPARISON THEOREMS FOR THE VOLUMES OF TUBES ABOUT METRIC BALLS IN CAT(𝜿)-SPACES

  • Lee, Doohann (Department of Mathematics, Sungkyunkwan University) ;
  • Kim, Yong-Il (School of Liberal Arts, Korea University of Technology and Education)
  • Received : 2011.04.08
  • Accepted : 2011.08.13
  • Published : 2011.09.30
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Abstract

In this paper, we establish some comparison theorems about volumes of tubes in metric spaces with nonpositive curvature. First we compare the Hausdorff measure of tube about a metric ball contained in an (n-1)-dimensional totally geodesic subspace of an n-dimensional locally compact, geodesically complete Hadamard space with Lebesgue measure of its corresponding tube in Euclidean space ${\mathbb{R}}^n$, and then develop the result to the case of an m-dimensional totally geodesic subspace for 1 < m < n with an additional condition. Also, we estimate the Hausdorff measure of the tube about a shortest curve in a metric space of curvature bounded above and below.

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References

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