Does a Riemannian manifold have a triangulation with quantitative bounds? Suppose that $M$ is a closed Riemannian manifold with bounded geometry, i.e., curvature between $-1$ and $1$ and injectivity radius at least $1$.  Since $M$ is a smooth manifold, it has a triangulation.  Does it necessarily have a triangulation that is "nice" with respect to the metric?
For instance, is there an $\epsilon>0$ such that for any such $M$, there is a triangulation of $M$ whose simplices are all homeomorphic to the standard simplex by a map $f$ such that $f$ and $f^{-1}$ are both $\epsilon^{-1}$-Lipschitz?
The method I'm familiar with for constructing a triangulation is to embed $M$ in $\mathbb{R}^n$, construct a fine net of points in $M$, construct the Delaunay triangulation of those points, then project back to the manifold to get a triangulation.  But this isn't very quantitative -- it depends on the embedding, and even if the embedding is nice, an unlucky choice of points will lead to some bad simplices.  Is there a better way?
 A: Let us use the following result (e.g., see Eichhorn, Global Analysis on open manfolds, Proposition 1.3)


*

*If $(M^m,g)$ is of bounded geometry (in the $C^\infty$-sense, say), then there 
exists $\epsilon_0 > 0$ such that for 
that for any $0<\epsilon <\epsilon_)$ there is a countable cover of $M$ by geodesic balls $B_{\epsilon}(x_j)$,
$\bigcup B_{\epsilon}(x_j) = M$, such that the cover of $M$ by the balls $B_{2\epsilon}(x_j)$ with double radius and the 
same centers is still uniformly locally finite.


Now choose points $y_k$ such that in each non-empty intersection of $B_{\epsilon}(x_j)$ they span an $m$-simplex, which is diffeomorphic to the standard $m$-simplex by any of the relevant exponential maps $\exp_{x_j}$. Then the maps you are looking for are just Riemannian exponential mappings which satisfy your assumptions by the properties of bounded geometry.  
A: A series of papers has been published (or rather is being published) on this topic, staring with Stability of Delaunay-type structures for manifolds (see http://dl.acm.org/citation.cfm?id=2261284) by Boissonnat, Dyer, and Ghosh. I think the paper in the series you might find most useful is Delaunay triangulation of manifolds (see https://arxiv.org/abs/1311.0117).  
