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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 where the number of solutions is generically finite)?

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)?