  Given a set of points in the Euclidean space $\mathbb{R}^\ell$ with $\ell>1$,
the pairwise distances between the points are determined by their spatial
location and the metric $d$ that we endow $\mathbb{R}^\ell$ with. Hence, the
distance $d(\mathbf x,\mathbf y)=\delta$ between two points is fixed by the
choice of $\mathbf x$ and $\mathbf y$ and $d$. We study the related problem of
fixing the value $\delta$, and the points $\mathbf x,\mathbf y$, and ask if
there is a topological metric $d$ that computes the desired distance $\delta$.
We demonstrate this problem to be solvable by constructing a metric to
simultaneously give desired pairwise distances between up to $O(\sqrt\ell)$
many points in $\mathbb{R}^\ell$. We then introduce the notion of an
$\varepsilon$-semimetric $\tilde{d}$ to formulate our main result: for all
$\varepsilon>0$, for all $m\geq 1$, for any choice of $m$ points $\mathbf
y_1,\ldots,\mathbf y_m\in\mathbb{R}^\ell$, and all chosen sets of values
$\{\delta_{ij}\geq 0: 1\leq i<j\leq m\}$, there exists an
$\varepsilon$-semimetric $\tilde{\delta}:\mathbb{R}^\ell\times
\mathbb{R}^\ell\to\mathbb{R}$ such that $\tilde{d}(\mathbf y_i,\mathbf
y_j)=\delta_{ij}$, i.e., the desired distances are accomplished, irrespectively
of the topology that the Euclidean or other norms would induce. We showcase our
results by using them to attack unsupervised learning algorithms, specifically
$k$-Means and density-based (DBSCAN) clustering algorithms. These have manifold
applications in artificial intelligence, and letting them run with externally
provided distance measures constructed in the way as shown here, can make
clustering algorithms produce results that are pre-determined and hence
malleable. This demonstrates that the results of clustering algorithms may not
generally be trustworthy, unless there is a standardized and fixed prescription
to use a specific distance function.
