Worst Packing Shapes


Yoav Kallus (Princeton University)


The question of which convex shapes leave the most empty space in their densest packing is the subject of various mathematical conjectures. We show that the ball is a local minimum of the optimal packing fraction in three dimensions among centrally symmetric shapes and the regular heptagon is a local minimum in two dimensions. In two dimensions and in dimensions above three the ball is not a local minimum, so the situation in three dimensions is exceptional despite what might be expected naively.

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