I'm currently working on a problem on the chapter of Geometry and Numbers from Andreescu and Dospinescu's Problems from the Book (highly recommended to read).
The problem statement is the following:
Consider a lamp (a point) in space. Prove that no matter how we place a finite number of closed spheres of equal radius, the light of this lamp will be able to go to infinity (that is, there exists a direction in which the light will not hit any of these spheres). The spheres must not touch. (Iran 2003)
The main result we are supposed to use is Minkowski's theorem:
Suppose $A\subseteq\mathbb R^n$ is a bounded, centrally symmetric, convex (and measurable) set with volume strictly bigger than $2^n$. Then $A$ contains a lattice point different from the origin.
First attempt
I originally interpreted the problem being in two dimensions and placed the lamp at the origin. Trying to blot out the light, I placed three spheres around it and then placed another 3 in the holes as follows:
This attempt led me to believe that I was not understanding the idea correctly, maybe light bounced off something or I wasn't quite seeing something. However I then understood (or maybe I'm still sidetracked and not seeing things correctly) that the problem was meant to be in 3D.
After the realization
Taking into account the third dimension, I want to proceed similarly to the first example in the chapter.
We suppose by contradiction that it's possible to blot out the light with a finite sphere packing. This tells us that somewhere out there is a packing of non-touching spheres which blots out the light.
If I was to use Minkowski's theorem I would like to find a set with volume greater than $8$ and in that set I would get a lattice point. Something tells me that the lattice point in question gives us the direction of the ray of light in which light escapes. However, the packing doesn't help to build our convex set.
At first, I thought that I should take the convex hull of the sphere packing but that might not work. That hull might not be centrally symmetric. And even if it did, finding a lattice point inside the convex hull doesn't guarantee that light won't touch any of the spheres.
My question
Is the way I'm proceeding a correct way to approach the problem?
Could you point me in the right direction in order to see the light? I believe that the issue with the problem is to find the correct solid and from there use the idea that the lattice point is the correct direction.
Also, if you know, can you recommend me similar problems to this one, where the statement is simple to read and we may use Minkowski's theorem to do them?
Thanks in advance for your help.
Your approach is plausible .. start by assuming a counterexample of a sphere arrangement that blots out the light. But it's not that difficult to find one, and once you've found it you have to conclude that the theorem is false.
I constructed a counterexample as follows with $14$ spheres of radius $1$. Two figures below give views of the construction.
The lamp, a point source, is at the origin $O=(0,0,0)$.
The first two spheres (blue in the figures below) are centered at $(\pm(1+\alpha),0,0), \alpha=.01$. These spheres blot out all light inside a double cone tangent to the two spheres and with apex at $O$. The spheres are a distance $2\alpha$ apart, and the remaining illumination forms a narrow region symmetric across the $yz$ plane. It's narrow enough to be blocked by two groups of $6$ spheres each.
The first group (red in the figures below) are centered on equally spaced points on a circle on the $yz$ plane of radius $2.1$. Thinking of the circle as a clock face, the centers are at $2,4,6,8,10,12$ o'clock.
A second circle on the $yz$ plane, centered at $O$ and with radius 3.6, contains the centers of the second group of spheres (green in the figures below). The centers are at $1,3,5,7,9,11$, and these spheres blot out the gaps left by the first group.
(There's nothing special about these two circle radius values $\dots$ a range of values would work, as long as the resulting spheres did not overlap and all the light was blocked.)
The $14$ spheres define $7$ double cones of shade, all with apex at $O$. Let $S$ be a sphere centered at $O$ large enough to contain the $14$ given spheres. The intersection of $S$ and the $7$ double cones consists of $14$ circles lying on $S$. I checked in Geogebra that these circles overlap and form a covering of $S$, so $S$ (a proxy for the sphere at infinity) is completely in the shade.
Below are two figures showing the spheres. The first is a view of the $yz$ plane. The second is a view of the $zx$ plane.
Conclusion
The statement of the theorem was already faulty, because it did not rule out that the lamp could be inside a sphere. But the counter example shows that the theorem is false. Nevertheless, it is tempting to speculate that no finite collection of spheres could block out all light if they all had to be separated by a minimum distance, such as, say $0.5$ of the sphere radii.