In 1990 W. Kuperberg conjectured that it is impossible to have seven infinite mutually disjoint unit cylinders all touching a unit sphere. As a first step towards a solution I would like to answer the following simpler question.
Let $L_1$, $L_2$, $L_3$ be three lines in $\mathbb{R}^3$, all tangent to the unit sphere $x^2+y^2+z^2=1$, with the additional property that the distance between any two of these lines is at least $1$. Let $M$ be an arbitrary point in $\mathbb{R}^3$.
Is it true that $$d(M, L_1)+d(M,L_2)+d(M,L_3)\ge 1+\frac{\sqrt{3}}{2}?$$
Here $d(M,L_i)$ denotes the Euclidean distance from point $M$ to line $L_i$.
Note that a much weaker lower bound of $\sqrt{3}$ can be easily proved even if the lines are not tangent to the unit sphere (but still at distance at least $1$ from each other).
I am totally stumped. In particular, is there a nice way to handle the condition that the lines are at least unit distance apart from each other?
Some hints We can assume one line fixed without loss of generality at say $(t,0,1), t\in \mathbb{R}$. Now investigate the possibilities for line two and three. Unallowed points to pass through will have a simple geometric shape based on line 1 ( it should be a filled cylinder, no? ). Or you could view it as you are trying to pack three cylinders of radius 0.5 each so their axes lay tangent to the sphere of radius 1 and they can also not intersect each other.