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Claim. The statement of Problem 1121 is true as the special case of a theorem for arbitrary convex bodies. Arseniy Akopyan, Alexey Balitskiy and Mikhail Grigorev, On the circle covering theorem by A. W. Goodman and R. E. Goodman, Discrete Comput. Geom. 59 (2018), no. 4, 1001--1009, define for a convex body $K\subset\mathbb R^d$ the parameter of asymmetry

$$\sigma=\min_{q\in\operatorname{int}K}\min\{\mu>0:(K-q)\subset-\mu(K-q)\}.$$

Their Theorem 2.1 states that a non-separable family of positive homothetic copies of $K$ with homothety coefficients $\tau_1,\ldots,\tau_n>0$ can always be covered by a translate of $\frac{\sigma+1}{2}(\sum_i\tau_i)K$. A centrally symmetric body has $\sigma=1$, so for the Euclidean disk with $\tau_i=r_i$ the theorem gives a covering disk of radius $\sum_ir_i$, which is the problem's statement. With the Minkowski and Radon bound $\sigma\le d$ (their Lemma 2.2) it gives the factor $\frac{d+1}{2}$ for every convex body, the corollary the abstract states, which improves Bezdek and Lángi's factor $d$.

The proof centers the homothet at $o=\sum_i\tau_io_i/\sum_i\tau_i$ and, if a point of the hull lay outside it, projects onto the direction orthogonal to a separating hyperplane, where Goodman and Goodman's segment lemma gives a contradiction. For symmetric bodies the paper credits this direct argument to F. Petrov, who in 2001 proposed the Euclidean case to the Open Mathematical Contest of Saint Petersburg Lyceum 239. This page follows the arXiv version of 16 February 2017.

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.