We have neither a proof nor a counter example for the following claim.
---------
"For N any positive integer, any convex polygon allows partitioning
into N *convex* pieces so that all the pieces have the same area *and*
same perimeter."
---------
The pieces need not be identical to one another. Only they should have
equal area and perimeter.

We could verify the following:
1. For some simple polygons (like a rectangle), there are multiple
ways of achieving a partition as required above.
2. For the special case N =2, the above claim holds.
3. If the pieces need not be convex, the equal area + equal perimeter
partitioning into N pieces is possible for any N - but we feel it is a
severely weakened result.

Note: If the claim above is valid, the following problem opens up: For a given convex polygon and number N, find that partitioning into convex pieces all with equal area and perimeter so that the total length of the 'cuts' is a minimum.

With regards,
R. Nandakumar
N. Ramana Rao