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?
