If the answer is ``Not always,'' 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?
There is work on partitioning convex polygons into equal area convex pieces so that every piece equally shares the boundary of the given target polygon: [ANRCU98] [AKK+98].
A proof of a weaker result--that any polygon allows fair partitioning for any n (where the pieces need not be convex) is proposed at http://nandacumar.blogspot.com/2006/10/cutting-shapes-ii.html.