Let $C$ be a right circular cone, the convex hull of a unit-radius disk and a point directly above the disk center at height $h$.

Is the most dense packing of $\mathbb{R}^3$ achieved by translates of $C$ and its negative,

^{*}$-C$, i.e., upside-down $C$? Or is it known that rotations of the cones permit achieving a denser packing?

^{*}
_{Barany, Imre, and Jiri Matousek. "Packing cones and their negatives in space." Discrete & Computational Geometry 38.2 (2007): 177-187. (Springer link)}

Here is

*TMA*'s idea:

(

*27Jun14*). Conjecture by Wlodek Kuperberg (see the comments),

*the*expert on the topic:

"For a generic cone, the densest packing by translates of the cone and its point-reflection is perhaps the best one."

Now marked as an open problem.

Dense packing of space with various convex solids, L. Fejes Toth Special Volume, Renyi Inst., Hungary, 2010; arxiv.org/pdf/1008.2398.pdf $\endgroup$ – Wlodek Kuperberg Jun 15 '14 at 1:52