The problem setup is simple: Given a $d$-simplex $\Delta_d:=\{(x_1,\cdots,x_d):x_i\geq 0,\sum_i x_i\leq 1\}$, can we find a finite sequence of parallelotope $A_i$ so that $\Delta_d=\cup_{i=1}^N A_i$, where $N$ may depend on $d$?

The problem in dimension $d=1,2$ is straightforward. However even in $d=3$, it is not clear to me if there is such a construction.

My main motivation for such a result is trying to reduce certain problem on simplex to parallelotope, which is easier to handle.

Any comment shall be greatly appreciated.

The problem setup is simple: Given a $d$-simplex $\Delta_d:=\{(x_1,\cdots,x_d):x_i\geq 0,\sum_i x_i\leq 1\}$, can we construct a finite sequence of parallelotope $A_i$ so that $\Delta_d=\cup_{i=1}^N A_i$, where $N$ may depend optimally on $d$ in the sense any sequence of such covers must have cardinality lower bounded by $N$ up to a universal constant.

The problem in dimension $d=1,2$ is straightforward. However even in $d=3$, it is not clear to me if there is such a construction.

My main motivation for such a result is trying to reduce certain problem on simplex to parallelotope, which is easier to handle.

Any comment shall be greatly appreciated.