We study all the ways that a given convex body in
dimensions can break into countably many pieces that move away from each
other rigidly at constant velocity, with no rotation or shearing. The initial
velocity field is locally constant a.e., but may be continuous and/or fail
to be integrable. For any choice of mass-velocity pairs for the pieces, such
a motion can be generated by the gradient of a convex potential that is
affine on each piece. We classify such potentials in terms of a countable
version of a theorem of Alexandrov for convex polytopes, and prove a stability
theorem. For bounded velocities, there is a bijection between the mass-velocity
data and optimal transport flows (Wasserstein geodesics) that are locally
incompressible.
Given any rigidly breaking velocity field that is the gradient of a continuous
potential, the convexity of the potential is established under any of several
conditions, such as the velocity field being continuous, the potential being
semiconvex, the mass measure generated by a convexified transport potential
being absolutely continuous, or there being a finite number of pieces. Also
we describe a number of curious and paradoxical examples having fractal
structure.
Keywords
least action, optimal transport, semiconvex functions,
power diagrams, Monge–Ampère measures