The most widely applied strategy for workload sharing is to equalize the workload assigned to each resource. In mobile multi-agent systems, this principle directly leads to equitable partitioning policies in which (i) the workspace is divided into subregions of equal measure, (ii) there is a bijective correspondence between agents and subregions, and (iii) each agent is responsible for service requests originating within its own subregion. In this paper, we design provably correct, spatially-distributed and adaptive policies that allow a team of agents to achieve a convex and equitable partition of a convex workspace, where each subregion has the same measure. We also consider the issue of achieving convex and equitable partitions where subregions have shapes similar to those of regular polygons. Our approach is related to the classic Lloyd algorithm, and exploits the unique features of power diagrams. We discuss possible applications to routing of vehicles in stochastic and dynamic environments. Simulation results are presented and discussed.