Question about "Cartesian product of p balls"

[dragon9001]

Dear Alex,

I am following your Hypobolic series work.
In this paper, I got lost when I met this concept,

My question is,
1) What does (D^n)^p mean?
2) How do we get that embedding? for example, the 50x2D, do we have to train 50 independent 2D Poincare embedding and then combine them?