  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. The topological ingredient is a Borsuk-Ulam
type statement on configuration space with the standard action of the symmetric
group. This result was discovered in increasing generality by Fuks, Vaseliev
and Karasev. We include a detailed proof and discuss how it relates to Gromov's
proof for the case p=2.
