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In this paper we show an equivalence relationship between additively weighted Voronoi cells in Rd, power diagrams in Rd and convex hulls of spheres in Rd. An immediate consequence of this equivalence relationship is a tight bound on the complexity of: (1) a single additively weighted Voronoi cell in dimension d; (2) the convex hull of a set of d-dimensional spheres. In particular, given a set of n spheres in dimension d, we show that the worst case complexity of both a single additively weighted Voronoi cell and the convex hull of the set of spheres is Θ(n[d/2]). The equivalence between additively weighted Voronoi cells and convex hulls of spheres permits us to compute a single additively weighted Voronoi cel1 in dimension d in worst case optimal time O(n log n+n[d/2]).