  Multidimensional scaling (MDS) is a popular technique for mapping a finite
metric space into a low-dimensional Euclidean space in a way that best
preserves pairwise distances. We study a notion of MDS on infinite metric
measure spaces, along with its optimality properties and goodness of fit. This
allows us to study the MDS embeddings of the geodesic circle $S^1$ into
$\mathbb{R}^m$ for all $m$, and to ask questions about the MDS embeddings of
the geodesic $n$-spheres $S^n$ into $\mathbb{R}^m$. Furthermore, we address
questions on convergence of MDS. For instance, if a sequence of metric measure
spaces converges to a fixed metric measure space $X$, then in what sense do the
MDS embeddings of these spaces converge to the MDS embedding of $X$?
Convergence is understood when each metric space in the sequence has the same
finite number of points, or when each metric space has a finite number of
points tending to infinity. We are also interested in notions of convergence
when each metric space in the sequence has an arbitrary (possibly infinite)
number of points.
