On Minimal Tilings with Convex Cells Each Containing a Unit Ball
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We investigate the following problem that one can regard as a very close relative 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 edge curvature of the cells? In particular, we prove that the average edge curvature in question is always at least \(13.8564\ldots\).
Key wordsTiling Convex cell Unit sphere packing Average edge curvature Foam problem
Subject Classifications05B40 05B45 52B60 52C17 52C22