Max of min and Min of max - 2

Reference: Earlier post

Let us define the distance between two convex regions as the haussdorf distance between them when the 2 regions are kept so as to minimize this hausdorff distance. Consider partitions of a convex region into n convex pieces such that (1) The average of the distance between the nC2 pairs of pieces is minimized or (2) the maximum among the nC2 such distances is minimized.

It seems conceivable (no reasons!) that minimizing the max pair distance need not necessarily make all pair distances equal. This needs some checking...

Will achieving such partitions optimizing the pair distances between pieces have any impact on the area/perimeter/diameter etc. of pieces?

Remark: I learned from Prof. Roman Karasev, that nC2 pair distances is a set of too many quantities to equalize. So, I guess the above thought may not be very useful.