Jian-Guo Liu, Duke University
Motivated by a study of least-action incompressible flows, we study all of the ways that a given convex body in Euclidean space can break into countably many pieces that move away from each other rigidly at constant velocity. Assuming they satisfy a least-action principle from optimal transport theory, we classify them in terms of a countable version of Minkowski's geometric problem of determining convex polytopes by their face areas and normals. A hypothesis that implies the least-action principle is the absence of mass concentrations in a certain multi-D flow that generalizes 1D sticky particle dynamics. Illustrations involve a number of curious examples both fractal and paradoxical, including Apollonian packings and other types of full packings by smooth balls.