  Sum-of-norms clustering is a popular convexification of $K$-means clustering.
We show that, if the dataset is made of a large number of independent random
variables distributed according to the uniform measure on the union of two
disjoint balls of unit radius, and if the balls are sufficiently close to one
another, then sum-of-norms clustering will typically fail to recover the
decomposition of the dataset into two clusters. As the dimension tends to
infinity, this happens even when the distance between the centers of the two
balls is taken to be as large as $2\sqrt{2}$. In order to show this, we
introduce and analyze a continuous version of sum-of-norms clustering, where
the dataset is replaced by a general measure. In particular, we state and prove
a local-global characterization of the clustering that seems to be new even in
the case of discrete datapoints.
