Consider the following basic model of finite crystal cluster formation: in a periodic graph G with vertices in R^d (representing possible molecular bonds) a subset (of atoms) must be chosen, so that the total number of bonds between a point in X and one outside X is minimized. These bonds form the edge-perimeter of X, denoted \partial X. If the graph is periodic and locally finite, any X satisfies an inequality of the form |X|^{d-1} \leq C |partial X|^d, where the optimal C depends on the graph. How can we determine the structure of sets X realizing equality in the above, based on the geometry and of G? If we take the continuum limit of G, then the classical Wulff shape theory describes optimal limit shapes, and at least two proofs of isoperimetric inequality apply, one based on PDE and calibration ideas, and the other based on Optimal Transport ideas. We focus on using the heuristic coming from the continuum analogue, to answer the above question in some cases, in the discrete case. This approach highlights the tight connection between discrete PDEs and semidiscrete Optimal Transport, and a link to the Minkowski theorem for convex polyhedra.

Let $\mu$ be a Borel probability measure on a compact path-connected metric space $(X, \rho)$ for which there exist constants $c,\beta\ge 1$ such that $\mu(B) \ge c r^{\beta}$ for every open ball $B\subset X$ of radius $r>0$. For a class of Lipschitz functions $\Phi:[0,\infty)\to\mathbb R$ that are piecewise within a finite-dimensional subspace of continuous functions, we prove under certain mild conditions on the metric $\rho$ and the measure $\mu$ that for each positive integer $N\ge 2$, and each $g\in L^\infty(X, d\mu)$ with $\|g\|_\infty=1$, there exist points $y_1, \ldots, y_{ N }\in X$ and real numbers $\lambda_1, \ldots, \lambda_{ N }$ such that for any $x\in X$, \begin{align*} & \left| \int_X \Phi (\rho (x, y)) g(y) \,\text{d} \mu (y) - \sum_{j = 1}^{ N } \lambda_j \Phi (\rho (x, y_j)) \right| \leqslant C N^{- \frac{1}{2} - \frac{3}{2\beta}} \sqrt{\log N}, \end{align*} where the constant $C>0$ is independent of $N$ and $g$. In the case when $X$ is the unit sphere $\mathbb{S}^{d}$ of $\mathbb R^{d+1}$ with the ususal geodesic distance, we also prove that the constant $C$ here is independent of the dimension $d$. Our estimates are better than those obtained from the standard Monte Carlo methods, which typically yield a weaker upper bound $N^{-\frac 12}\sqrt{\log N}$.