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Claim. The statement of Problem 1121 is true as the special case of a more general inequality. Hugo Hadwiger, Nonseparable convex systems, Amer. Math. Monthly 54 (1947), no. 10, 583--585, calls a system of convex curves $C_1,\ldots,C_n$ in the plane nonseparable when no line misses every curve and has curves on both of its sides, and proves, for such a system with perimeters $L_i$, diameters $D_i$ and circumradii $R_i$ and with $C_0$ the boundary of its convex hull, that

$$L_0\le\sum_{i=1}^nL_i,\qquad D_0\le\sum_{i=1}^nD_i,\qquad R_0\le\sum_{i=1}^nR_i .$$

For circles the circumradius of $C_i$ is its radius $r_i$, and the circumscribed disk of the convex hull contains every disk of the family, so the third inequality says that nonseparable disks lie in one disk of radius $\sum_i r_i$, which is the problem's statement. The paper follows Goodman and Goodman's and cites it; the site's curator records it as a proof of a generalization, not an independent proof.

Its statement is taken from the zbMATH review of the printed article (Zbl 0030.22004) and from the site's page; this page does not reproduce the proof or determine how far it reuses Goodman and Goodman's argument.

Depends on. No page of this wiki.

Acceptance. The result is refereed: it appeared in the American Mathematical Monthly. The site's curator, Thomas Bloom, marks the problem proved and records on its page that Hadwiger proved a generalization to convex bodies (problem page last edited 17 April 2026). The problem was first settled by Goodman and Goodman, whose page carries the Lean formalization the site flags.