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As we know,if M is an Alexandrov space with sec>=1,then the cone over M has sec>=0.What if when M is a cuboid with side length r1,...,rn,dia(M)<=π,then the cuvature of the cone over M bounded below by what?And the case when M is a ball in a hyperbolic space with dia(M)<=π?

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In this case there is no lower bound near the tip of the cone. –  Anton Petrunin Apr 27 '13 at 5:23
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