Intersecting cylinders around a sphere Intersecting $n$ unit-radius cylinders, each with axis through the origin,
produces a shape circumscribed about a unit-radius sphere:
   
My question is:

For each $n$, which arrangement of cylinders minimizes the
Pompeiu–Hausdorff distance
to the sphere?

This Hausdorff distance is
the smallest $r$ such that each set is contained within an
$r$-neighborhood of the other.
For the above $2$-cylinder example, I believe the Hausdorff distance is
$\sqrt{2}-1$.
For $n=3$, there are two natural candidates:
orthogonal cylinder axes, and axes in a plane at $60^\circ$:

I calculate the Hausdorff distances to the sphere to be
$\sqrt{\frac{3}{2}}-1$
and $\frac{2}{\sqrt{3}}-1$ respectively,
approximately $0.22$ and $0.15$.
If I'm not miscalculating,
it is a bit of a surprise that the second arrangement is closer to the sphere.
 A: According to RavenclawPrefect's answer, we need to choose the unit vectors $a_1,a_2,\dots,a_n$ so as to minimize the maximal value of
$$
  f(v)=\min\langle v,a_i\rangle,
$$
over $v\in S^2$. If $\cos \alpha$ is the answer to this optimizational problem, then the sets
$$
  Z_i=\{v\in S^2\colon \langle v,a_i\rangle\leq\cos\alpha\}
$$
cover the sphere.
The set $Z_i$ is a zone on the sphere: a `strip' of angular width $2\alpha$. Here I use the terminology from this nice paper by Jiang and Polyanskii.
In that paper, the authors show that, if $n$ zones cover the sphere, then the sum of their widths is at least $\pi$, thus establishing Fejes Toth's conjecture. Thus, if the (equal) zones corresponding to $a_1,\dots,a_n$ cover the sphere, their widths should be at least $\pi/n$. So, for any tuple $(a_1,a_2,\dots,a_n)$ there is a vector whose angle with each of them is at most $\pi/2-\pi/(2n)$.
This shows that the $n$ cylinders rotated by equal angles aroung the same axis indeed provide an optimal arrangement.
A: Place a pin through a cylindrical straw, passing through the axis and perpendicular thereto. When you revolve the straw around the pin, you trace a sphere with the straw contained entirely outside.
Distributing $n$ cylinders at equal angles around such an axis will give a good approximation to this spherical envelope.
