hyperbolic geometry - Volume of a geodesic simplex on a manifold of non-positive curvature.

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 simplic...Read more