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We raise and investigate the following problems that one can regard as very close relatives of the densest sphere packing problem. If the Euclidean 3-space is partitioned into convex cells each containing a unit ball, how should the shapes of the cells be designed to minimize the average surface area (resp., average edge curvature) of the cells? In particular, we prove that the average surface area (resp., average edge curvature) in question is always at least 24/√3 = 13.8564.... This estimate is improved further for Voronoi tilings of unit ball packings.