I am asking this question to know more 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 that can 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 is encountered in general area in maths (for example coding theory)? What are the known developments?