We known that Croke proved a isoperimetric inequality for the four dimensional Cartan-Hadamard manifold,I want to ask extend to Alexandrov space that whether have same isoperimetric inequality ?Whether may be added to some condition,Is $CD\left ( n,k \right )$ (curvature dimension),Is $MCP\left ( n,k \right )$(measure contraction property) and create to New condition?Of course,Maybe inequality is same.
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$\begingroup$ You definitely need some extra conditions, but I do not see why you mention CD(n,k) and MCP(n,k). Maybe the right condition is extendability of geodesics; but even if it is true, it will not be easy. $\endgroup$– Anton PetruninCommented Sep 1, 2012 at 4:08
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1$\begingroup$ CD and MCP are in the wrong direction: they are curvature lower bounds, while Croke's theorem is for manifolds of curvature bounded above by $0$. What would make sense would be to consider CAT(0) spaces. Then you may have trouble defining perimeter and volume in a way that enables the problem to be phrased properly. I do not know a result in this setting. $\endgroup$– Benoît KloecknerCommented Sep 1, 2012 at 10:08
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