Title: Worst Packing Shapes

Author (Talk): Yoav Kallus, Princeton University

Abstract:

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 unusual despite what might be expected naively.