Given a finite metric space $X$, the doubling constant of $X$ is the smallest integer $k$ such that any ball of arbitrary radius $r$ can be covered by at most $k$ balls of radius $r/2$. The doubling dimension of $X$ is defined to be $\dim X = \log_2 k$.
Despite references to doubling dimension being replete among computer science paper abstracts, and especially within computational geometry and topology, it is mostly assumed that the doubling dimension is known a priori.
As far as computation of doubling dimension, in "Local Doubling Dimension of Point Sets" there is the remark on page $3$:
It is $\mathrm{NP}$-hard to calculate the doubling dimension of a metric [$8$] but it can be approximated within a constant factor [$9$, Sec. $9$].
In [8], however, the problem is explained thusly:
Interestingly, the problem of computing the exact doubling dimension of a point set is NP-hard. (This result seems to be folklore.) Yet this fact has not deterred the development of algorithms that are based on the doubling dimension, partly because it can be approximated within a constant factor, and partly because many of these algorithms function without explicit knowledge of the doubling dimension—it appears only in the analysis.
My question is: Is the $\mathrm{NP}$-hardness of exact doubling dimension computation "folklore" because it is trivial, or because it is really hard to prove? If the former, does anyone know a proof? If the latter, what is the heuristic by which one concludes it is probably $\mathrm{NP}$-hard?
Edit: After further Googling, I discovered there is a presentation by the authors of [8] above, apparently titled "It is $\mathrm{NP}$-hard to determine the doubling dimension of a set S" given at this link, which unfortunately does not seem to work without some type of authorization.
