On a conjecture by Eckhoff and Dolnikov concerning line transversals to Euclidean disks

Let $K$ be a convex body in the Euclidean plane ${\mathbb{R}}^2$. We say that a point set $X\subseteq {\mathbb{R}}^2$ satsfies the property $T(K)$ if the family of translates $\{K+x:x\in X\}$ has a line transversal. A weaker property, $T(K,s)$, of the set $X$ is that every subset $Y\subseteq X$ consisting of at most $s$ elements satisfies the property $T(K)$. The following question goes back to Gr"unbaum: given $K$ and $s$, what is the minimal positive number $\lambda =\lambda (K,s)$ such that every finite point set in ${\mathbb{R}}^2$ with the property $T(K,s)$ also satisfies the property $T(\lambda K)$? The constant ${\lambda }_{disj}(K,s)$ is defined similarly, with the only additional assumption that the translates $x+K$ and $y+K$ are disjoint for every $x,y\in X$, $x\mathrm{\ne }y$. One case of particular interest is $s=3$ and $K=B$, where $B$ is a unit Euclidean ball. Namely, it was conjectured by Eckhoff and, independently, Dolnikov that $\lambda (B,3)=\frac{1+\sqrt{5}}{2}$. In this paper we propose a stronger conjecture, which, on the other hand, admits an algebraic formulation in a finite alphabet. We verify our conjecture numerically on a sufficiently dense grid in the space of parameters and thereby obtain an estimate ${\lambda }_{disj}(B,3)\le \lambda (B,3)\le 1.645$. This is an improvement on the previously known upper bounds $\lambda(B, 3) \leq \frac{1 + \sqrt{1 + 4\sqrt{2}}}{2} \approx 1.79$ (Jer'onimo Castro and Rold'an-Pensado, 2011) and ${\lambda }_{disj}(B,3)\le 1.65$ (Heppes, 2005).