We consider a variational problem in a planar convex domain, motivated by statistical mechanics of crystal growth in a saturated solution. The minimizers are constructed explicitly and are completely characterized.
Based on a construction by Kashiwara and Rouquier, we present an analogue of the Beilinson-Bernstein localization theorem for hypertoric varieties. In this case, sheaves of differential operators are replaced by sheaves of W-algebras. As a special case, our result gives a localization theorem for rational Cherednik algebras associated to cyclic groups.