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Claim. The statement of Problem 1121 is true as the special case of a theorem for centrally symmetric bodies. Károly Bezdek and Zsolt Lángi, On non-separable families of positive homothetic convex bodies, Discrete Comput. Geom. 56 (2016), no. 3, 802--813, call a family $\mathcal K=\{x_i+\tau_iK_0: x_i\in\mathbb R^d,\ \tau_i>0,\ i=1,\ldots,n\}$ non-separable when no hyperplane disjoint from $\bigcup\mathcal K$ strictly separates some members from the others, and write $\lambda(\mathcal K)$ for the least $\lambda>0$ such that a translate of $\lambda(\sum_i\tau_i)K_0$ covers $\bigcup\mathcal K$. Their Theorem 4 (Section 5) states that for every o-symmetric convex body $K_0$ and every non-separable family $\mathcal K$ of its positive homothets, $\lambda(\mathcal K)\le 1$, for all $d\ge2$ and $n\ge2$. With $K_0$ the Euclidean unit disk and $\tau_i=r_i$ this is the problem's statement; the case $n=1$ is trivial. The abstract states the result for balls of any norm on $\mathbb R^d$ and names it as Erdős's conjecture, proved for the Euclidean norm by Goodman and Goodman.

The proof is a variant of Goodman and Goodman's. A strengthened form of their segment lemma (Lemma 3) shows that intervals $[x_i-\tau_i,x_i+\tau_i]$ whose union is an interval are covered by the interval of half-length $\sum_i\tau_i$ about $\sum_i\tau_ix_i/\sum_i\tau_i$. Projecting the family orthogonally onto any line through the origin gives such intervals, so the support function of the hull is bounded by that of $x+(\sum_i\tau_i)K_0$, with $x$ the $\tau$-weighted mean of the centers. The same paper gives counterexamples, families of triangles, to Goodman and Goodman's conjecture that the bound holds for every convex body, and proves $\lambda(\mathcal K)\le d$ in general. This page follows the arXiv version of 13 May 2016.

Depends on. No page of this wiki.

Acceptance. The result is refereed: it appeared in Discrete and Computational Geometry. The site's page does not mention the paper.