Let $M$ be a compact Riemannian manifold. I'll call a set of non-empty subsets $C\_1,\dots,C\_N$ a *geodesic tiling* of $M$ if:


* Each $C\_n$ is closed (geodesically) convex hull of a finite number of $\{p\_{i,n}\}\_{i=1}^{k\_n}$ in $C\_n$; i.e. the smallest geodesically convex subset of $M$ containing the points $\{p\_{i,n}\}\_{i=1}^{k\_n}$,
* $\cup\_{n=1}^N\, C\_n=M$
* If $n\neq n'$ then $C\_n$ and $C\_{n'}$ have disjoint interiors.


It seems to me intuitive that such a collection of sets $\{C\_n\}\_{n=1}^N$ must exist. But does it? And where can I find a reference?



Do all compact manifolds admit geodesic tiling
If dimension $=2$, then "yes".


If dimension $\ge 3$, then "no".
If such a tiling would exist, then the common boundary of a pair of $C$'s would contain a geodesic hypersurface, but generic Riemannian manifold does not have such hypersurfaces.


For more on convex hulls in Riemannian world, see [our paper](https://arxiv.org/abs/2103.15189).


