Problem 1121

If $C_1,\ldots,C_n$ are circles in $\mathbb{R}^2$ with radii $r_1,\ldots,r_n$ such that no line disjoint from all the circles divides them into two non-empty sets then the circles can be covered by a circle of radius $r=\sum r_i$ .

Source: erdosproblems.com/1121

Geometry

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SolvedProved

An accepted solution exists. The statement is true.

PROVED (LEAN): the site credits Goodman and Goodman's 1945 proof, records Hadwiger's 1947 generalization and Bezdek and Litvak's 2016 alternative proof, and flags a Lean formalization; see the claim pages.

BeLi16: Bezdek, Károly and Litvak, Alexander E., Packing convex bodies by cylinders. Discrete Comput. Geom. (2016), 725-738.
GoGo45: Goodman, A. W. and Goodman, R. E., A circle covering theorem. Amer. Math. Monthly (1945), 494-498.
Ha47: Hadwiger, H., Nonseparable convex systems. Amer. Math. Monthly 54 (1947), no. 10, 583-585. The site's commentary cites [Ha47] for Hadwiger's generalization, and the formal-conjectures statement file lists this paper under the key, but the site's reference record resolves the key to Marshall Hall's Cyclic projective planes, Duke Math. J. 14 (1947), 1079-1090, a paper on another problem; Hadwiger's paper is the one the commentary means.
ABG18: Akopyan, A., Balitskiy, A. and Grigorev, M., On the circle covering theorem by A. W. Goodman and R. E. Goodman. Discrete Comput. Geom. 59 (2018), no. 4, 1001-1009; arXiv:1605.04300.
BeLa16: Bezdek, K. and Lángi, Z., On non-separable families of positive homothetic convex bodies. Discrete Comput. Geom. 56 (2016), no. 3, 802-813; arXiv:1602.01020.

Problem 1121

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