'Spectrums' of convex regions of same area and perimeter

This is a bit of speculation that takes off from this discussion.
Consider all planar convex regions of same area and perimeter. As was noted over several mathoverflow discussions, one can define the distance between any pair of them - C1 and C2 - in various ways:

(1) area of intersection region when C1 is placed over C2 in such a way as to maximize this area (2) perimeter of their intersection when this intersection perimeter is maximized (3) the Hausdorff distance between them when C2 is kept over C1 so as to minimize this distance (4) the least average closest distance over the boundary of one of them to the to the other. And so on...

Question: Which (if any) of these distance measures can cause the planar regions all of same area and perimeter to line up in a spectrum with a specific order? What we mean by this lining up is: if C0 and C1 are the specific pair of regions that are farthest from each other (call these extreme configs) and if Ci and Cj are two other regions, then if distance (C0,Ci) < distance (C0, Cj), it automatically should imply that distance (Ci, C1) > distance(Cj, C1) - for that will give the order C0-Ci-Cj-C1.

Note: One add other quantities to the set being kept equal over the convex regions that are being lined up - diameter for instance.

Guess: An ellipse is a good candidate for an extreme configuration - that is, one of C0, C1 is an ellipse.