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Let$M$ be a Complete simply-connected n-dimensional Riemannian manifold with nonpositive curvature,$\Omega $is a open subset of $M$ ,If $n\geq 4$Is anyone give a estimate of $\frac{Vol_{n}\left ( \Omega \right )}{Vol_{n-1}\left ( \Omega \right )^{\frac{n}{n-1}}}$ ?,If $\Omega $replace by $B^{n}\left ( 1 \right )$ `,it is a unit sphere,,estimate?

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I guess you want to say $$\frac{Vol_{n}\left ( \Omega \right )}{Vol_{n-1}\left (\partial \Omega \right )^{\frac{n}{n-1}}}.$$ –  Anton Petrunin Aug 31 '12 at 12:20

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It is an old open problem.

  • The case $n=4$ is done by Croke. He shows that $$\frac{Vol_{n}\left ( \Omega \right )}{Vol_{n-1}\left (\partial \Omega \right )^{\frac{n}{n-1}}}\le C_n$$ for some constant $C_n$ which is optimal for $n=4$.
  • The case $n=3$ is done by Kleiner.
  • The cases $n=1$ and $2$ are trivial.
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I would not say that $n=2$ is trivial, even if it is not difficult. It can be proved in several ways, but the first proof by Weil in 1926 can be considered to have launched the subject, except that many people seem to have forgotten this work. –  Benoît Kloeckner Sep 11 '12 at 21:17

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