We consider the problem of balancing the load among several service-providing facilities, while keeping the total cost low. Let D be the underlying demand region, and let p1,... ,pm be m points representing m facilities. We consider the following problem: Subdivide D into m equal-area regions R1.,..., Rm, so that region R i is served by facility pi, and the average distance between a point q in D and the facility that serves q is minimal. We present constant-factor approximation algorithms for this problem, with the additional requirement that the resulting regions must be convex. As an intermediate result we show how to partition a convex polygon into m = 2k equal-area convex subregions so that the fatness of the resulting regions is within a constant factor of the fatness of the original polygon. We also prove that our partition is, up to a constant factor, the best one can get if one's goal is to maximize the fatness of the least fat subregion. We also discuss the structure of the optimal partition for the aforementioned load balancing problem: indeed, we argue that it is always induced by an additive-weighted Voronoi diagram for an appropriate choice of weights.