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?