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VOLUME ESTIMATE VIA TOTAL CURVATURE IN HYPERBOLIC SPACES

Published online by Cambridge University Press:  20 March 2003

ALBERT BORBÉLY
Affiliation:
Kuwait University, Department of Mathematics and Computer Science, P.O. Box 5969, Safat 13060, Kuwaitborbely@mcs.sci.kuniv.edu.kw
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Abstract

Let $D\subset H^n(-k^2)$ be a convex compact subset of the hyperbolic space $H^n(-k^2)$ with non-empty interior and smooth boundary. It is shown that the volume of D can be estimated by the total curvature of $\partial D$. More precisely, $(n-1)k^n{\rm Vol}(D)+ {\rm Vol}(S^{n-1})\leq \int_{\partial D}K$, where K denotes the Gauss–Kronecker curvature of $\partial D$ and Vol$(S^{n-1})$?> denotes the Euclidean volume of the sphere.

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NOTES AND PAPERS
Copyright
© The London Mathematical Society 2003

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Footnotes

This research was supported by Kuwait University Research Grant SM 03/2000.