Erdős Problem #1121
PROVED LEANdatabase updated 2026-04-19
The statement as posed is true, and a proof exists. A solution has also been checked by machine, in Lean.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
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 \(r=\sum r_i\). This is true, and was proved by Goodman and Goodman [GoGo45] (whose proof also generalises to higher dimensions). A generalisation to convex bodies was proved by Hadwiger [Ha47]. An alternative proof is given by Bezdek and Litvak [BeLi16].
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 1 further variant / known partial result in the Lean file. The authoritative wording is on erdosproblems.com/1121.Prizenone
Topicsgeometry
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/1121
Attempts
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