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Statement

Setting (p. 13). $c_{\mathbb Z}(n,d)$ is the largest contact number of a packing of $n$ balls of unit diameter in $\mathbb E^d$ whose centers are points of the integer lattice $\mathbb Z^d$.

Theorem 7.8 (p. 13). For all $n>1$ and $d\ge2$, $c_{\mathbb Z}(n,d)\le\lfloor dn-dn^{\frac{d-1}{d}}\rfloor$.

The survey says (p. 14), citing Bezdek, Szalkai and Szalkai (its reference [15], Discrete Math. 339 (2015)) and Theorem 6.1, that the bound is sharp for $d=2$ and every $n>1$ and for $d\ge3$ and every $n=k^d$ with $k>1$, and that it is not sharp for $d=3$, $n=5$.

Proof pointer

Pp. 13--14, recalled from the cited paper. The unit cubes centered at the $n$ lattice points form a box-polytope whose surface volume is $2dn-2c_{\mathbb Z}(n,d)$. Lemma 7.9, proved on pp. 13--14 from the Brunn--Minkowski inequality, says cubes have the least surface volume among box-polytopes of given volume; through Corollary 7.10 this gives $2dn-2c_{\mathbb Z}(n,d)\ge2dn^{(d-1)/d}$.

Read depth

Claims checked: the definition, the theorem and its proof were read on the page images of the print. The sharpness statements are cited from the 2015 paper and were not checked. Nothing here is independently reviewed.

Dependencies

None in the corpus.

Source. K. Bezdek and M. A. Khan, Contact numbers for sphere packings, in New Trends in Intuitive Geometry, Bolyai Society Mathematical Studies, Springer (2018), 25--47, doi:10.1007/978-3-662-57413-3_2; the label and page are those of arXiv:1601.00145v2, the edition read, named on the source card.

Bears on

• Problem 1084: points of $\mathbb Z^d$ are at mutual distance at least $1$ and their touching pairs are the pairs at distance $1$, so $c_{\mathbb Z}(n,d)\le f_d(n)$. The theorem bounds only these lattice configurations. With the sharpness the survey reports at $n=k^d$, $k>1$, it gives $f_d(k^d)\ge c_{\mathbb Z}(k^d,d)=dk^d-dk^{d-1}$ for $d\ge2$.