co.combinatorics – Trade-off between radius and covering number using $d$-dimensional balls


Given any $d$-dimensional solid $X$, let the length of the longest line segment connecting two points of $X$ be equal to $1$. How can we prove the following conjecture?

For any integer $n ge 1$, there exists a radius $r$ and a positive constant $c$ (i.e., independent of $X$, $n$, $d$ and $r$) such that

$$rlefrac{c}{n^{1/d}}$$

where $n$ is the number of $d$-dimensional balls having radius $r$ that completely cover $X$, with possible overlaps.


If the conjecture is false in general, it is possible to add a condition bounding $d$ (as a function of $n$ or $r$) such that the inequality holds?