Fair Partitioning of Convex Polygons

Define a fair partitioning of a polygon as a partition of it into a finite number of pieces so that every piece has both the same area and the same perimeter. If all the resulting pieces are convex, call it a fair convex partitioning. Given any positive integer $n$, can any convex polygon be convex fair partitioned into $n$ pieces?

If the answer is \textquotedblleft{}Not always,\textquotedblright{} how does one decide the possibility of such a partitioning for a given polygon and a given $n$? And if a fair convex partition exists for a specific polygon, how does one find a fair partitioning that minimizes the total length of the cut segments, or minimizes the sum of the perimeters of the pieces?

And finally, what could one say about higher dimensional analogs of this question?