The 3-ball is a local pessimum for packing (Q400987)

\textit{M. Gardner}, in his [New mathematical diversions. Rev. ed. Washington, DC: MAA, Mathematical Association of America (1995; Zbl 0842.00006)] published a 1972 conjecture of Ulam that the sphere was the ``worst case of dense packing of identical convex solids''. The paper under review considers the problem for lattice packings (by translates) of centrally symmetric bodies.NEWLINENEWLINEThe symmetry of the Euclidean ball means that no packing of that body can improve on the best packing with translates. It also means that its lattice packings have strictly fewer degrees of freedom than do those of any other body of the same dimension. One might, as the author points out, reasonably expect this to make it a pessimal packer in every dimension. \textit{K. Reinhardt}, however, showed in [Abh. Math. Semin. Univ. Hamb. 10, 216--230 (1934; Zbl 0009.32003; JFM 60.0536.13)] that this is not so in the plane; the regular octagon does worse, as do shapes arbitrarily Hausdorff-close to the disc [\textit{K. Mahler}, Proc. Akad. Wet. Amsterdam 50, 108--118 (1947; Zbl 0036.31002)].NEWLINENEWLINEA body is called \textit{irreducible} if any proper compact subset packs more tightly (the Euclidean disc is a simple example). It is called \textit{globally} (resp. \textit{locally}) \textit{pessimal} if its tightest packing is looser than that of any other body (resp. any other body that is sufficiently Hausdorff-close). As mentioned above, the Euclidean disc has neither of these latter properties.NEWLINENEWLINEThe author shows (using geometric lattice theory) that the unit \(n\)-ball is irreducible but not locally pessimal for \(n=4,5\), and reducible for \(n=6,7,8,24\). He shows, using spherical harmonics, that it is locally pessimal for \(n=3\). NEWLINENEWLINENEWLINEDespite the advanced methods used, this paper is readable by the non-specialist, which is good, as the result is one that should be of wide interest.