[Math] Cutting convex sets

Question

Any bounded convex set of the Euclidean plane can be cut into two convex pieces of equal area and circumference.

Can one cut every bounded convex set of the Euclidean plane into an arbitrary number $n$
of convex pieces having equal area and circumference?

The solution of this problem for $n=2$ is generically unique. Are there other values of $n$
(assuming that the problem is possible) where this happens?

More generally, given a $d-$dimensional bounded convex set $C$ in the Euclidean space $\mathbf E^d$ of dimension $d$. for which values of $n$ can one cut $C$ into $n$ convex pieces of equal area with boundaries of equal $(d-1)-$dimensional area? ($n=2$ is again easy, but the solution is no longer generically unique if $d>2$). Are there values for $(n,d)$ for which the solution always exists in a generically unique way (or for which the number of solutions is generically finite)?

Answer 1

Permit me (as a late-comer) to add a bit more information. The 2D version of your question was posed by R. Nandakumar and N. Ramana Rao and posted at TOPP. They have written "an introduction" to their "Fair Partitioning" problem: http://arxiv.org/abs/0812.2241. They cite a forthcoming paper by Barany, Blagojevic, and Szucs that settles the problem (positively) for $n=3$.

---

Addendum (3Dec10): A significant advance on this problem was just announced by Boirs Aronov and Alfredo Hubard , the latter of whom is giving a seminar talk on this at NYU. They solved it (positively) for all prime powers $n$. Here is the abstract of the talk, entitled "From the sandwich to the waist":

I will talk about a Ham Sandwich/Borsuk Ulam type theorem. Let $K$ be a convex body, for any prime power $n=p^k$ it is possible to partition $K$ into $n$ convex pieces with equal areas and equal perimeter.

This confirms a conjecture of Nandakumar and Ramana Rao (for all prime powers). The proof uses some basic ideas from optimal transport and from equivariant topology. It turns out that this is closely related to one of the main ingredients of the proof by Gromov of the Waist of the Sphere inequality.

Here is a paper on Gromov's "waist of the sphere" theorem.

Edit. Here is an arxiv paper by Aronov & Hubard on their work.