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Statement

Setting (pp. 1--2, 10). $\mathbf K$ is a convex body (a compact convex set with non-empty interior) in $\mathbb E^d$, and $c(\mathbf K,n,d)$ is the largest contact number of a packing of $n$ translates of $\mathbf K$. For $d\ge3$: $\delta(\mathbf K)$ is the density of a densest packing of translates of $\mathbf K$; $\mathrm{iq}(\mathbf K)=\mathrm{svol}_{d-1}(\mathrm{bd}\,\mathbf K)^d/\mathrm{vol}_d(\mathbf K)^{d-1}$ is its isoperimetric quotient; the Hadwiger number $H(\mathbf K)$ is the largest number of non-overlapping translates of $\mathbf K$ that all touch $\mathbf K$; the one-sided Hadwiger number $h(\mathbf K)$ is the largest number of non-overlapping translates of $\mathbf K$ that touch $\mathbf K$ and all lie in a closed supporting halfspace of $\mathbf K$; and $\mathbf K_{\mathbf o}=\frac12(\mathbf K+(-\mathbf K))$.

Theorem 7.1 (p. 10). For every convex body $\mathbf K$ in $\mathbb E^d$, $d\ge3$,

$$c(\mathbf K,n,d)\le\frac{H(\mathbf K_{\mathbf o})}{2}\,n-\frac{1}{2^d\,\delta(\mathbf K_{\mathbf o})^{\frac{d-1}{d}}}\sqrt[d]{\frac{\mathrm{iq}(\mathbf B^d)}{\mathrm{iq}(\mathbf K_{\mathbf o})}}\;n^{\frac{d-1}{d}}-\bigl(H(\mathbf K_{\mathbf o})-h(\mathbf K_{\mathbf o})-1\bigr) \le\frac{3^d-1}{2}\,n-\frac{\sqrt[d]{\omega_d}}{2^{d+1}}\,n^{\frac{d-1}{d}},$$

where $\omega_d=\pi^{d/2}/\Gamma(\frac d2+1)=\mathrm{vol}_d(\mathbf B^d)$.

The survey attributes the theorem to Bezdek (its reference [9], J. Combin. Theory Ser. A 98 (2002), 192--200). It recalls (p. 10) Hadwiger's bound $H(\mathbf K)\le3^d-1$ and the bound $h(\mathbf K)\le2\cdot3^{d-1}-1$, each with equality exactly for affine $d$-cubes.

Proof pointer

Not proved in the survey. It names (p. 11) Theorem 7.3, a density bound for the union of the doubled translates $\mathbf c_i+2\mathbf K_{\mathbf o}$, as playing an important role in the published proof.

Read depth

Claims checked: the definitions and both inequalities were read on the page images of the print. The cited proof was not read. Nothing here is independently reviewed.

Dependencies

None in the corpus. External input: Bezdek (2002), which the survey cites for the proof.

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: through its case $\mathbf K=\mathbf B^d$, Corollary 7.2, an upper bound for $f_d(n)$ when $d\ge3$.