Convex Equipartitions of volume and surface area.

We show that, for any prime power $p^k$ and any convex body $K$ (...) in $\mathbb{R}^d$, there exists a partition of $K$ into $p^k$ convex sets with equal volume and equal surface area.

I believe the (or at least one) source here is a MathOverflow question by Nandakumar.