Global maximality of the FCC lattice for truncated density of unit ball packings
Let
be a packing of unit balls in $^3$, and letdenote its
-truncated density for $>0$. The corresponding FCC lattice is the face-centred cubic lattice packing of unit balls in $^3$. **FCC truncated-density conjecture.** For all $>0$, among packings of unit balls in $^3$, the-truncated density has a maximum at the corresponding FCC lattice.
The preceding discussion notes that the analogous statement for lattice packings has a local maximum at the FCC lattice. The conjecture asks for global maximality among all packings; no resolution is supplied in the source.
References
Primary source
Károly Bezdek and Zsolt Lángi, “Density bounds for unit ball packings relative to their outer parallel domains”, arXiv:2401.00645 (2024).
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