TITLE: Cutting convex sets
QUESTION [19 upvotes]: 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)?

REPLY [2 votes]: There is a generalization of this question, where "area" and "circumference" are replaced by arbitrary "nice" measures (for the purpose of this answer, say absolutely continuous measures) $\mu$ and $\nu$ on $\mathbb{R}^2$. Bárány and Matoušek have a nice paper on the subject. 
Even more generally, fix nice probability measures $\mu_1,\ldots,\mu_i$ on $\mathbb{R}^2$. 
A $k$-fan in $\mathbb{R}^2$ consists of $k$ rays (semi-infinite lines) $r_1,\ldots,r_k$ emanating from a point, listed in some clockwise order. (In fact $k$-fans are also allowed to emanate from the point at infinity, i.e., a set of $k$ parallel lines is considered to be a $k$-fan.) Write $C_k$ for the region proceeding $r_k$ in the clockwise order. 
Given a vector $\alpha=(\alpha_1,\ldots,\alpha_k)$ with non-negative entries summing to one, 
say that $\mu_1,\ldots,\mu_r$ can be simultaneously $\alpha$-partitioned if 
there exists a $k$-fan such that $\mu_i(C_j)=\alpha_j$ for each $i=1,\ldots,r$ and $j=1,\ldots,k$. (If $\alpha_1=\ldots=\alpha_k=1/k$ say that the measures can be simultaneously equipartitioned. This case, with $k=2$, is closest to the original )
Bárány and Matoušek have a whole host of results about when such partitions exist and do not exist. Here are just a couple: 

For any $k \geq 5$ and any $\alpha$, there are two measures that can not be simultaneously $\alpha$-partitioned. 
For any $\alpha=(\alpha_1,\alpha_2)$, any two measures can be simultaneously $\alpha$-partitioned, even if the center of the fan is specified in advance. 

No one knows, for example, if any two measures can be simultaneously equipartitioned into four parts. Karasev seems to have a paper where he proves that any two measures can be simultaneously equipartitioned into $q$ convex parts, whenever the number of parts is a prime power. (This was first  achieved for three parts -- this is the result by Bárány et al that Joseph O'Rourke mentioned.) I am unclear on the relation between this and the result of Hubard and Aronov, mentioned by Joseph O'Rourke in his answer.
Higher-dimensional versions have also been considered but much is open. For example, for any three measures in $\mathbb{R}^3$ can one always find a convex $3$-partition of space so that each measure has measure $1/3$ on each part? (I heard Bárány say in a seminar that the version with $3$ replaced by a power of $2$, is known to be true; but I didn't note down a reference.)