Convex Equipartitions of volume and surface area

Understand

We show that, for any prime power p^k and any convex body K (i.e., a compact convex set with interior) in Rd, there exists a partition of K into p^k convex sets with equal volume and equal surface area.

• We derive this result from a more general one for absolutely continuous measures and continuous functionals on the space of convex bodies.
• This result was independently found by Roman Karasev generalizing work of Gromov and Memarian who proved it for the standard measure on the sphere and p=2.
• Our proof uses basics from the theory of optimal transport and equivariant topology.