This is a follow-up question to [this][1].

Since it is not always possible to construct such partition, I would like to know if there are additional restrictions which we could impose so that the wanted partition exists in the $n$-dimensional setting ($n\geq 1$).


  [1]: https://mathoverflow.net/questions/369421/is-it-always-possible-to-partition-a-b-timesc-d-into-disjoint-blocks-d-i