Suppose that we divide some disk into $n$ parts of equal area in such a way that we choose some point $A_1$ on the circle and then find $(n-1)$ points $A_2,...A_n$ on the circle in such a way that the lines $A_1A_2,...,A_1A_n$ divide the disk into $n$ parts of equal area.
Now take some closed convex set $C$ which is not a disk and transform the disk with bijective holomorphic mapping onto $C$.
Now generally the lines $A_1A_2,...A_1A_n$ will be transformed into curves $C_1C_2,...C_1C_n$.
Does there exist $C$ such that curves $C_1C_2,...,C_1C_n$ divide $C$ into $n$ parts of equal area no matter where the position of $A_1$ is?