$\DeclareMathOperator{\vol}{Vol}$ Suppose we are working inside a Riemannian $n$-manifold $M$ of non-negative Ricci curvature. In his PhD thesis (see pp.8–9), Zhong-dong Liu presents an incredibly simple method for constructing coverings of subsets of $M$ of diameter $d$ by closed balls of radius $\mu d/2$, where the number of balls in the covering is bounded above by a constant that depends only on $\mu$ and $n$. His construction relies solely on the following special case of the Bishop-Gromov inequality: $$\frac{\vol(B(p,R))}{\vol(B(p,r))}\leq\frac{R^n}{r^n},$$ where $p$ is any point in $M$, and $0<r<R$. This means that his construction is valid in any metric space where the above inequality makes sense and is valid, such as an $n$-dimensional Alexandrov space of non-negative curvature. Since his idea is so simple, and yet so relatively recent (1991) and applicable to such a large class of objects, I find it very hard to believe that he was the first to make it. Does anyone know of an earlier appearance of this simple application of the Bishop-Gromov inequality to this ball covering problem? I want to be honest and cite the argument, as I did not come up with it myself; however, I don't want to cite what may be a classical idea as though it were novel.
$\begingroup$
$\endgroup$
2
-
$\begingroup$ This is essentially Gromov's compactness theorem (which is Bishop--Gromov inequality was invented for) --- see "Structures métriques pour les variétés riemanniennes" by Gromov (published in 1981). $\endgroup$– Anton PetruninDec 16, 2020 at 20:51
-
$\begingroup$ Thanks for this—I think I've found the relevant section, Ch.5 A. I guess it's somewhere between Definition 5.1 and End of Proof of 5.3. I can't quite see where this ball covering is done though, as it all seems far more complicated than what Liu writes. Do you know where exactly he constructs the covering? (I'm reading the expanded English version, but I think the section is question is relatively unchanged). $\endgroup$– Tom SharpeDec 17, 2020 at 16:27
Add a comment
|