ABSTRACT

We study all 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 without shear.  If originated by a convex velocity potential, we characterize such flows by a countable version of Minkowski's geometric problem of determining convex polytopes by their face areas and normals.  We provide several criteria ensuring convexity of the potential in locally rigid breaking flows.  This leads to a classification of all least-action incompressible flows from a convex bounded domain in optimal transport theory.  Illustrations involve a number of curious examples both fractal and paradoxical, including Apollonian packings and other types of full packings by smooth balls.