/sci/ - Science & Math

I have n convex sets in the euclidean plane, I take the first and second, a and b respectively, and rotate and translate them over each other such that their intersection is maximized, now their union is a new set, call it (ab), we take the third set c and rotate and translate it such that it's intersection with set (ab) is maximized, this new set we call (abc), we take the fourth set and do the same process with the previously generated set, and we do this with all n sets, and end up with the set(abc... n) which within itself contains all sets from a to n, the question is is the set (abc... n) the same as the set that has minimum area but also contains any set from a to n?