  We prove an asymptotic crystallization result in two dimensions for a class
of nonlocal particle systems. To be precise, we consider the best approximation
with respect to the 2-Wasserstein metric of a given absolutely continuous
probability measure $f \mathrm{d} x$ by a discrete probability measure $\sum_i
m_i \delta_{z_i}$, subject to a constraint on the particle sizes $m_i$. The
locations $z_i$ of the particles, their sizes $m_i$, and the number of
particles are all unknowns of the problem. We study a one-parameter family of
constraints. This is an example of an optimal location problem (or an optimal
sampling or quantization problem) and it has applications in economics, signal
compression, and numerical integration. We establish the asymptotic minimum
value of the (rescaled) approximation error as the number of particles goes to
infinity. In particular, we show that for the constrained best approximation of
the Lebesgue measure by a discrete measure, the discrete measure whose support
is a triangular lattice is asymptotically optimal. In addition, we prove an
analogous crystallization result for a problem where the constraint is replaced
by a penalization. These results can also be viewed as the asymptotic
optimality of the hexagonal tiling for an optimal partitioning problem. They
generalise the crystallization result of Bourne, Peletier and Theil
(Communications in Mathematical Physics, 2014) from a single particle system to
a class of particle systems, and prove a case of a conjecture by Bouchitt\'{e},
Jimenez and Mahadevan (Journal de Math\'ematiques Pures et Appliqu\'ees, 2011).
