Partitioning Convex 2D Regions - Moment of Inertia

 After the original 'fair partition' question of partitioning a 2D convex region C into n convex pieces all of same area and perimeter, one recently worried (http://nandacumar.blogspot.com/2020/12/partition-into-equal-width-and-equal.html) about questions such as whether trying to minimize the max perimeter of n convex pieces would automatically lead to a partition where all pieces have same perimeter.

Then one mentioned instead of perimeter, quantities such as diameter, least width of pieces that could be used as the basis of the convex partition.

Now let me mention one more such quantity which might be of interest - the Moment of Inertia of the pieces with the MI of each piece measured with reference to an axis normal to the plane thru the center of mass of the piece.  One could try to (1) make each convex piece have same moment of inertia  or (2) try to  maximize the minimum (or minimize the maximum) of this quantity among the pieces. 

Another motivation for thinking about so many different quantities - area, perimeter, diameter, least width and now the moment of inertia: Let us restrict our consideration only to polygonal 2D regions. Then, there is a hope that with values specified for a limited set of such global quantities (one might also need to  specify the number of sides n the polygonal region has), one could fully determine a convex polygonal region for any n. By 'limited set', one means a set with number of elements considerably less than O(n).