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The classical Besicovitch covering lemma (BCL) asserts that for any $d \geq 1$, there is a constant $N(d)$ with the following property. If $A \subset \mathbb{R}^d$ is any subset and $r : A \to (0,R]$ is a (bounded) function ($R < \infty$ is fixed), then there are (at most) countably many points $\{a_j\}_j$ such that $\{B(a_j, r(a_j))\}_j$ is a cover of $A$ with multiplicity at most $N(d)$ (i.e. for any $j$, $B(a_j, r_j)$ has nonempty intersection with at most $N(d) - 1$ balls $B(a_k, r_k)$, $k \neq j$).

I should underscore the fact that the the multiplicity $N(d)$ depends only on $d$ and on none of: the 'ball size function' $r$, the bound $R$, or the set $A$.

I am interested in a generalization of this fact to manifolds, replacing $\mathbb{R}^d$ with a (finite dimensional, smooth, Riemannian) manifold $M$ and Euclidean balls with geodesic balls. Unfortunately, it seems that the BCL is generally false in this setting, due to a result of Chi: indeed, according to that paper, if the BCL holds on a Hadamard manifold (i.e. everywhere nonpositive sectional curvature), then that manifold is $\mathbb{R}^d$ for some $d$.

So, one must compromise to obtain a positive result. I think that it suffices to constrain the maximum ball size $R$: given a manifold $M$, the BCL holds for ball size functions $r \leq R$, where $R$ depends on $M$. For e.g., one can apply the Nash embedding theorem and the classical BCL, and this implies the following: for any compact $M$, there is are numbers $N = N(M)$ and $R = R(M) > 0$ such that for any $A \subset M$ and any $r : A \to (0,R]$, there is an at-most countable cover $B(a_j, r(a_j)$ of $A$ by geodesic balls with multiplicity $\leq N$.

Question: Is there any way to tell what $R$ should be by looking at only local features of the manifold?

The constraint $R$ comes from how 'folded up' the Nash/Whitney embedding is in $\mathbb{R}^d$, and I have no clue how to control it. So, this result is highly unsatisfying: for small enough $R$, the BCL is a local result (isn't it?), and so the multiplicity shouldn't have anything at all to do with the global 'folding up' of an embedding in $\mathbb{R}^d$. Instead, I suspect that one ought to be able to cook up a constraint on $R$ that depends only on some local feature, like the sectional or Ricci curvatures. Has anyone done this or know of prior work in this direction?

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In the Federer's book "Geometric Measure Theory", there is a notion of "directionally limited" metric space. He proves that the Besicovitch Covering Lemma holds for the directionally limited spaces.

It is easy to see that lower bound on sectional curvature plus upper bounds on dimension and diameter imply that the space is directionally limited. (Instead of diameter you may bound $R$.)

The same can be proved by applying the proof of the Besicovitch Covering Lemma; almost no modifications.

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  • $\begingroup$ This is exactly what I was looking for. Very satisfying to my intuition, that one must bound the number of 'directions' at every point: this is analogous to bounding the cardinality of a 'satellite' configuration as in the literature on the best possible constant $N(d)$ for the BCL. $\endgroup$ – A Blumenthal May 22 '14 at 0:47

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