TITLE: Hiding $k$ disks inside a larger disk
QUESTION [6 upvotes]: Suppose one has $k$ unit-radius disks, and the goal is to hide them inside
a disk of radius $R \gg k$.
The detection probes are rays along a line.
(Think of the disks as tumor cells, and the rays as radiation.)

Q1. What is the optimal hiding configuration for $k$ disks?

That is, how can the $k$ disks be arranged to be difficult to detect
by ray probes?
I believe the probability of detection from, say, random ray probes,
would be proportional to the integral, over all directions $\theta \in [0,\pi)$,
of the measure of the projection/shadow of the disks in direction $\theta$.
So I seek a configuration with the smallest mean shadow.
For example, it seems that for $k=3$, the obvious is the optimal configuration:

 
 
 


But it is unclear to me if the optimal configuration under this projection-based integral measure is identical to, say, (a) the optimal packing of $k$ disks in a surrounding disk of minimal radius, or (b) the packing of $k$ disks with the minimum area convex hull.

Q2. Is the projection integral measure identical to any of the well-known,
  previously studied disk packing constraints?

Or perhaps what I am suggesting has already been studied on its own, in which case
a pointer would be appreciated—thanks!

REPLY [2 votes]: Q2: If the union of the discs is conected, or more generaly, if no line separates the discs in a nontrivial way, then the integral of the lengths of the projections over $[0, \pi)$ is equal to the perimeter of the convex hull of the union of the discs. See, for example, Lemma 1 in this paper on opaque sets, where a related problem is studied.
It can be proved by approximating the perimeter by a polygonal curve and integrating each edge separately.