Let $M$ be a simply connected manifold that admits a metric of non-positive curvature. For example take $\mathbb{R}^k\times \mathbb{H}^n$. Take $m+1$ points $x_0$, $x_1, \ldots$ $x_m$, $m>k+1$ and consider the geodesic simplex $\sigma$ spanned by these points: that is, connect $x_0$ and $x_1$ by a geodesic segment, connect $x_2$ to every point on that geodesic and so on. The following estimate seem to hold $$vol(\sigma)\le C \text{ diam}(\sigma)^{k+1} $$ Questions: Is this optimal? What are the methods for estimating volume of geodesic simplices through geometry of the manifold? Maybe there are nice bounds that use some characteristics other than diameter?

Here is the basic setup: $M=\mathbb{R}^k\times \mathbb{H}^n$. If $m\le k+1$ then clearly the estimate above is optimal in $k$. However if $m>k+1$ then you cannot put your simplex into a flat. I have little Riemannian intuition, so it's not clear to me that one cannot do better, say $$vol(\sigma)\le C \text{ diam}(\sigma)^{k+1-\varepsilon} $$ Another question is the following. Assume that the restriction of the curvature to the simplex is $\le -0.1$. It seems to me that volume can still be unbounded, correct? Can one get better estimates?

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    $\begingroup$ In what sense do you mean optimal? There are some manifolds where the growth rate of the maximum volume of a simplex with diameter $d$ grows as a smaller function, e.g., there is a constant upper bound in hyperbolic space. $\endgroup$ – Douglas Zare Jul 13 '12 at 22:30
  • $\begingroup$ To add to @Doug's comment: to get a better bound you might want to supply a lower bound on the curvature. $\endgroup$ – Igor Rivin Jul 14 '12 at 9:24

The magic words are "cone type inequality". For a quite nice discussion of this and related subject (with copious references) see Wenger's thesis.


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