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This question is a follow-up of the following negative question.

Let $(X,d)$ be a (non-empty) compact metric space.

More generally than in the first post, 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 and geodesically-convex
    • (but not necessarily the geodesically convex hull of a finite number of points),
  • $\cup_{n=1}^N\, C_n=X$
  • If $n\neq n'$ then $C_n$ and $C_{n'}$ have disjoint interiors.

When does $(X,d)$ admit such a cover?

If the question is too general, what about when $(X,d)$ is a compact subset of Euclidean space?

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  • $\begingroup$ The Euclidean space can be subdivided into cubes, it induces the needed subdivision for $X$. (Or, did you want to use induced intrinsic metric?) $\endgroup$
    – Anton Petrunin
    Commented Aug 1, 2022 at 21:38
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    $\begingroup$ The space $X$ must have a lot of geodesic hypersurfaces, but it is hard to make precise what is "a lot". Among sufficient conditions: $X$ is Riemannian and has constant sectional curvature. $\endgroup$
    – Anton Petrunin
    Commented Aug 1, 2022 at 21:39
  • $\begingroup$ @AntonPetrunin For the Riemannian case, if I'm allowed to take any other metric generating the same topology as the innducws intrinsic metric (possibly non-equivalent metrics) is there hope in dimension above 3? $\endgroup$
    – ABIM
    Commented Aug 4, 2022 at 5:06

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