Given any $d$-dimensional shape $X$ in the Euclidean space, let $\ell(X)$ be the length of the longest line segment connecting two points of $X$. How can we prove the following statement?
There exists a positive constant $c$ such that, for all integers $n \ge 1$, we can completely cover any $d$-dimensional shape $X$ with $n$-many $d$-dimensional balls having radius $r\le c\frac{\ell(X)}{n^{1/d}}~.$
Note: I can prove the above inequality by combining the bound of the question Trade-off between hypervolume and diameter of $d$-dimensional shapes having a hypercubic smallest bounding box with the well known bound $n\le\left(\frac{3}{r}\right)^d\cdot\frac{V(X)}{V(B)}$, where $V(\cdot)$ is the volume of a shape, and $B$ is the unit norm ball. I am wondering whether it is possible to avoid the use of the inequality of my previous question proving the above statement in a simpler and more direct way.