About a problem of fitting a cube in a subset of a sphere I am asking this question to know more about this problem that I find very interesting. 
The problem is that suppose you have the unit 2-sphere $S^2$ in $\mathbb{R}^3$ and a measurable subset $A \subset S^2$ such that $\mu(A)=0.9\mu(S^2)$. Then prove that you can find a cube whose vertices will fit inside the set $A$.
This question has been asked and answered before:
https://math.stackexchange.com/questions/573926/surface-of-a-sphere-and-cube
https://math.stackexchange.com/questions/499854/problem-regarding-the-fitting-cube-into-sphere
I want to know where this question originates from. Is this question part of some general type of questions that are encountered in a more general setting (for example coding theory)? What are the known developments?
 A: I asked a more general version of this question in an earlier question, 
Regions on a sphere that avoid a fixed point set.

          


          

A $5$-point set, its convex hull (blue) and smallest enclosing circle (red).


The OP's version is an $8$-point set. 
An upperbound was proved:
"for an $n$-point set $P$,
the area of an avoiding set cannot be larger
than $\frac{n-1}{n} A$, where $A$ is the area of the sphere."
For $n=8$, this gives an upper bound of $\frac{7}{8}=0.875$ which is smaller
than the OP's $0.9$.

To address the OP's specific questions:

I want to know where this question originates from. 

My version was original, but ...

Is this question part of some general type of questions that are encountered in a more general setting?

Yes, avoiding sets. For example, there is quite a bit of literature on mutually avoiding sets, going back to Erdős. There is also considerable literature on sets avoiding integral (or rational) distances. See, e.g.,
the question Integer-distance sets. Or:
Kurz, Sascha, and Valery Mishkin. "Open sets avoiding integral distances." Discrete & Computational Geometry 50.1 (2013): 99-123.
