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:
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![Disks3Projection][1]
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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&mdash;thanks!

  [1]: https://i.sstatic.net/Y3zTJ.jpg