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Claim. The statement of Problem 1121 is true. Károly Bezdek and Alexander E. Litvak, Packing convex bodies by cylinders, Discrete Comput. Geom. 55 (2016), no. 3, 725--738, posted as arXiv:1507.05115 on 17 July 2015, treat the problem in their last section. Following Hadwiger, a finite family of closed disks in the plane is nonseparable when no line disjoint from all of them divides them into two nonempty sets; its convex hull $K$ is called an NS-domain and the sum of the diameters its NS-diameter $\operatorname{diam}_{NS}(K)$. Theorem 6.1 states that planks forming an $r$-fold packing in an NS-domain $K$ have total width at most $r\operatorname{diam}_{NS}(K)$, and that the circumradius $R_K$ of $K$ satisfies

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

with equality exactly when $2R_K=\operatorname{diam}_{NS}(K)=\operatorname{diam}(K)$. The displayed inequality is the problem's statement: the circumradius of the hull of nonseparable disks of radii $r_1,\ldots,r_n$ is at most $\sum_i r_i$, so the circumscribed disk of the hull covers the family. The proof is analytic. Lemma 6.3 shows that a nonnegative integrable function on a convex body $K$ whose integral over every hyperplane section of the interior is at least $\Delta$ has total integral at least $2\Delta R_K$, by placing the weighted centroid at the origin and taking moments in the direction of a farthest point; summing Falconer's extremal functions of the generating disks gives a function with section integrals at least $1$ and total integral $\operatorname{diam}_{NS}(K)$, and the two bounds combine. The paper points to Goodman and Goodman for a completely different proof of the same inequality.

This page follows the arXiv version of 21 November 2015, not the journal text. The source card bezdek_2016_packing_convex_bodies_cylinders pages Theorem 6.1 on p. 10 and Lemma 6.3 on p. 11 of that version.

Depends on. No page of this wiki.

Acceptance. The result is refereed: it appeared in Discrete and Computational Geometry. The site's curator, Thomas Bloom, marks the problem proved and records on its page that Bezdek and Litvak give an alternative proof (problem page last edited 17 April 2026). The problem was first settled by Goodman and Goodman; this page records a different proof of the same statement.