Ulam's packing conjecture

The conjecture says that the optimal density for packing congruent spheres is smaller than that for any other convex body.

That is, according to the conjecture, the ball is the convex solid which forces the largest fraction of space to remain empty in its optimal packing structure.

Yoav Kallus has shown that at least among point-symmetric bodies, the ball constitutes a local maximum of the fraction of empty space forced.

It is conjectured that regular heptagons force the largest fraction of the plane to remain uncovered.

[4] In dimensions above four (excluding 8 and 24), the situation is complicated by the fact that the analogs of the Kepler conjecture remain open.