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Statement

Setting (pp. 3, 4, 9--10). A plank in $\mathbb R^d$ is a $(d-1)$-codimensional cylinder; in the plane it is the region between two parallel lines, and its width is the Euclidean distance between them. $r$-fold packings are as in Definition 4.1 (p. 4): each plank's base lies in the corresponding projection of $K$, and each point of $K$ lies in the interiors of at most $r$ of the truncated planks. Following Hadwiger, a finite family of closed circular disks in $\mathbb R^2$ is separable when some line disjoint from all the disks divides the plane into two open half-planes each containing at least one disk; otherwise it is a non-separable arrangement, an NS-family. The convex hull of an NS-family is an NS-domain $K$, and the sum of the diameters of its disks is its NS-diameter $\operatorname{diam}_{NS}(K)$. $R_K$ is the circumradius of $K$ (the radius of the smallest disk containing $K$) and $\operatorname{diam}(K)$ its Euclidean diameter.

Theorem 6.1 (p. 10). Let $K$ be an NS-domain in $\mathbb R^2$. If finitely many planks form an $r$-fold packing in $K$, then the sum of their widths is at most $r\operatorname{diam}_{NS}(K)$. Moreover

$$2R_K\le\operatorname{diam}_{NS}(K), \qquad (15)$$

with equality if and only if $2R_K=\operatorname{diam}_{NS}(K)=\operatorname{diam}(K)$.

The paper presents the theorem (p. 10) as improving Theorem 5.1 with $c_2=1$ for NS-domains with $\operatorname{diam}_{NS}(K)=2R_K=\operatorname{diam}(K)$.

Proof pointer

Pp. 12--13. For each plank $C_i$ take a unit normal $u_i$ and the ridge function $\frac1r\chi_{[a_i,b_i]}(\langle x,u_i\rangle)$, where $[a_i,b_i]$ is the range of $\langle x,u_i\rangle$ over $C_i$. Lemma 6.2 (p. 11), applied to these functions under the $r$-fold packing, gives $\sum_iw(C_i)\le r\,m(\mathcal L_1^+(K))$, inequality (13), with $m(\mathcal L_\Delta^+(K))$ the least integral of a nonnegative function on $K$ whose integral over every line meeting the interior of $K$ is at least $\Delta$ (see Lemma 6.3). Summing, over the disks of the family, Falconer's extremal function for a disk of radius $R$, $\frac1{\pi R}(R^2-|x|^2)^{-1/2}$ on the open disk with $m(\mathcal L_1^+(RB_2^2))=2R$ (Theorem 1.2 of Falconer's paper), gives $m(\mathcal L_1^+(K))\le\operatorname{diam}_{NS}(K)$, inequality (14). This with (13) gives the plank bound, and Lemma 6.3 with (14) gives (15). The paper calls the equality case straightforward and does not write it out, and it refers to Goodman and Goodman for a different proof of (15).

Read depth

Claims checked: the definitions, Theorem 6.1, Lemma 6.2 and the proof on pp. 12--13 were read clause by clause on the print. The equality case is asserted without proof in the paper and was not checked; Falconer's Theorem 1.2 is cited, not proved, and was not read.

Dependencies

Lemma 6.3. External input: K. J. Falconer, Function space topologies defined by sectional integrals and applications to an extremal problem, Math. Proc. Cambridge Philos. Soc. 87 (1980), 81--96, Theorem 1.2.

Source. K. Bezdek and A. E. Litvak, Packing convex bodies by cylinders, Discrete Comput. Geom. 55 (2016), no. 3, 725--738, doi:10.1007/s00454-016-9760-z; labels and pages are those of arXiv:1507.05115v2 (21 November 2015), as the source card records.

Bears on

• Problem 1121: the problem's hypothesis, that no line disjoint from all the circles divides them into two nonempty sets, is the paper's NS-family condition, and the diameters of the disks sum to twice the sum $\sum r_i$ of their radii. Inequality (15) therefore puts the family inside a disk of radius $R_K\le\sum r_i$, and so inside the concentric disk of radius $\sum r_i$, which is the problem's statement. The paper refers to Goodman and Goodman for a completely different proof of (15).