Lines Tangent to Four Unit Balls

Definitions and mathematical statement

Let $B_i=\{x\in{\mathbb{R}}^3:\|x-c_i\|\leq1\}$ for $1\leq i\leq n$; overlaps are explicitly permitted. Count geometric unoriented affine lines once. A line $\ell$ is tangent to $B_i$ when $\operatorname{dist}(c_i,\ell)=1$. Under the closed-ball meaning of "miss," an admissible line has an index set $I\subset\{1,\ldots,n\}$ with $|I|=4$ such that \[ \operatorname{dist}(c_i,\ell)=1\quad(i\in I),\qquad \operatorname{dist}(c_j,\ell)>1\quad(j\notin I). \] Thus it touches exactly four balls and meets no other closed ball. If missing only interiors was intended, additional tangencies and counting multiplicities would need separate treatment.

Counting problem. Determine the number, in the worst case as a function of $n$, of lines tangent to four unit balls and missing all the others. With an intended admissible configuration class $\mathcal C_n$, its extremal form is \[ M(n)=\sup_{(B_1,\ldots,B_n)\in\mathcal C_n} \#\{\ell:\ell\text{ satisfies the conditions above}\}. \] No general-position or isolated-tangent hypothesis is stated. This is essential for a finite combinatorial count: four distinct collinear centers admit a continuous family of parallel common tangent lines, and with $n=4$ there are no other balls to exclude them. The intended restriction, or a convention counting only isolated lines, must be recovered rather than silently added.

The maintained-entry evidence and summary report the problem open and conjecture $\Omega(n^3)$. For a finite-count interpretation that lower-bound statement would mean configurations with at least $cn^3$ admissible lines for all sufficiently large $n$ and a fixed $c>0$; it is not a conjectured upper bound by itself. The summary also mentions up to $12$ common tangents per quadruple, without the nondegeneracy conditions or citation; that finite per-quadruple claim cannot cover arbitrary continuous families. No cubic construction or matching upper bound is asserted here.

Short English statement

Find the worst-case number of lines touching exactly four unit balls in three dimensions and missing all remaining balls, with overlapping balls allowed. The source conjectures a cubic lower bound. Missing general-position or isolated-line conventions are crucial because degenerate configurations can have infinitely many common tangents.

Sources

• Local UnsolvedMath registry, permanent ID 5500061, source code catalogue entry, "Lines Tangent to Four Unit Balls"; supplied status open, classification OPEN-TRIAGE, updated 2026-08-06.
• The Open Problems Project, Computational Geometry, Problem 61, https://topp.openproblem.net/p61. The record reports source-TeX extraction and access on 2026-07-29; the maintained entry is described as "Open, conjectured to be $\Omega(n^3)$."
• Public source specification: https://github.com/edemaine/topp. The repository LICENSE is reported to scope software expressly; problem-text reuse remains NEEDS_REVIEW.
• The summary mentions a $12$-tangent per-four-sphere result but gives no publication, hypotheses, or theorem reference.