Abstract  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 arek disjoint convex sets we give a compact representation of the Voronoi diagram, usingO (k) line segments, that is sufficient for logarithmic time post-office location queries and motion planning. If these sets are polygons withn total vertices given in standard representations, we compute this diagram optimally in Θ (k logn) deterministic time for the Euclidean metric and inO (k logn logm) deterministic time for the convex distance function defined by a convexm-gon.