A Compact Piecewise-Linear Voronoi Diagram for Convex sites in the Plane

In the plane, the post-office problem, which asks for the closest site to a query site, and retraction motion planning, which asks for a one-dimensional retract of the free space of a robot, are both classically solved by computing a Voronoi diagram. When the sites are k disjoint convex sets, we give a compact representation of the Voronoi diagram, using O(k) line segments, that is sufficient for logarithmic time post-office location queries and motion planning. If these sets are polygons with n total vertices given in standard representations, we compute this diagram optimally in O(k log n) deterministic time for the Euclidean metric and in O(k log n log m) deterministic time for the convex distance function defined by a convex m-gon.