Optimality of Greedy Circle Packings

Definitions and mathematical statement

Let $B_{\pm}=\{(x,y)\in{\mathbb{R}}^2:(x\mp1)^2+(y-1)^2\leq1\}$ be the two fixed unit disks. The closed curvilinear triangular region between these disks and the $x$-axis is \[ T=\left\{(x,y)\in{\mathbb{R}}^2:-1\leq x\leq1,\quad 0\leq y\leq1,\quad (x-1)^2+(y-1)^2\geq1,\quad (x+1)^2+(y-1)^2\geq1\right\}. \] For $a\in{\mathbb{R}}$ and $r>0$, write \[ D(a,r)=\{(x,y)\in{\mathbb{R}}^2:(x-a)^2+(y-r)^2\leq r^2\}. \] This disk is tangent to the $x$-axis from above. An admissible packing is a countably infinite family $\mathcal P=(D(a_n,r_n))_{n\in{\mathbb{N}}}$, with every $r_n>0$, such that \[ D(a_n,r_n)\subseteq T\quad(n\in{\mathbb{N}}),\qquad \operatorname{int}D(a_n,r_n)\cap\operatorname{int}D(a_m,r_m)=\varnothing\quad(n\ne m). \] Thus boundary tangencies are permitted. Define its total area and the optimal supremum by \[ A(\mathcal P)=\pi\sum_{n\in{\mathbb{N}}}r_n^2, \qquad M=\sup\{A(\mathcal P):\mathcal P\text{ is an admissible packing}\}. \] The fixed bounding disks $B_{\pm}$ are not included in this area sum.

Source question. Does the greedy procedure described as successively inserting a disk tangent to the $x$-axis and to two previously placed disks produce an admissible infinite packing $\mathcal G$ satisfying \[ A(\mathcal G)=M? \] The equality asks for actual attainment of the supremum by the resulting infinite packing.

Unresolved procedural qualification. The supplied record does not fully define this greedy procedure. In particular, it does not explicitly specify the initial configuration, whether the two fixed bounding disks count as previously placed disks, how the two disks are selected at each insertion, or which gaps must eventually be filled. Counting the bounding disks as initial disks and recursively filling the gaps between neighboring tangent disks is a possible interpretation, but it is not explicitly stipulated in the supplied text. Consequently, the variational target above is precise, while the particular greedy packing whose optimality is being asked about requires clarification.

Short English statement

Among all infinite packings of disks tangent to the $x$-axis inside the gap bounded by the two specified unit disks, maximize the sum of the disk areas. The question asks whether the intended greedy insertion procedure attains this supremum. Its initialization and disk-selection rules are not fully specified in the supplied record.

Sources

• Local UnsolvedMath registry, permanent ID $30002821$, title Optimality of Greedy Circle Packings, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2015 and supplies no individual proposer attribution.
• Supplied report citation: Discrete Differential Geometry (2015), Oberwolfach Reports, pp. 661--729, DOI 10.4171/owr/2015/13. Supplied source and literature-source link, labeled "OWR Report 13/2015": https://doi.org/10.4171/owr/2015/13. The collection describes these reports as records of mathematical workshops and their open-problem discussions; no problem-specific page is supplied here.
• Supplied literature assessment, checked on exactly 2026-08-21: the record labels the problem open and states that global area optimality of the greedy two-neighbor insertion rule remains unestablished and that no authoritative counterexample was found. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment. The record itself requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review of its cleanup. These are supplied editorial assertions, not mathematical verification of this draft or of greedy optimality.