The Convex Geometry of Integrator Reach Sets

We study the convex geometry of the forward reach sets for integrator dynamics in finite dimensions with bounded control. We derive closed-form expressions for the volume and the diameter (i.e., maximal width) of these sets in terms of the state space dimension, control bound, and time. These results are novel, and use convex analysis to give an analytical handle on the "size" of the integrator reach set. Several examples are provided to illustrate our results. We envision that the ideas presented here will motivate further theoretical and algorithmic development in reach set computation.