Optimal configs of planar convex bodies based on some integrals.

Recording a bunch of questions on convex geometry...

We consider relative positions of 2 given planar convex bodies C1 and C2 optimizing the 4 quanitities described below. In the following, by C1 or C2, we mean boundary of C1 or C2. And we are allowed to translate, rotate, flip both bodies (subcases are got by disallowing some of these transformations):

(1) minimize the integral over C1 of the distance of a variable point P on C1 to the closest point to it on C2.
(2) minimize the integral over C1 of distance of a variable point P on C1 to the farthest point to it on C2.
(3) and (4) minimize the two integrals above with C1 and C2 interchanged.

Questions:
- For any specified C1 and C2 will all the 4 different optimizations always give the same configuration (relative position) of C1 and C2 relative to each other? As examples given below show, all 4 optimizations need not give the same relative position of C1 and C2. Then, the natural question is: will all 4 optimizations give different configs in general? One can also ask for {C1, C2} pairs for which the optimal configs are most different (one needs to define difference suitably then).
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Note: It can be seen that the centers of mass of C1 and C2 (viewed as uniform laminae) need not be coincident in say, the optimal config that minimizes the integral over C1 of distance to closest point on C2. Indeed, let C1 be a square and C2, an identical square with a segment of a circle added - this circle segment is formed by a very high radius arc passing thru 2 adjacent vertices of the square; now, to minimize the integral, we should put the square portion of C2 directly over C1 (which is of course, just a square) and then, the CMs of C1 and C2 won't coincide.
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- Will any of these optimizations also automatically achieve other optimizations such as say, maximize the area (or perimeter) of the intersection between C1 and C2? Indeed, in example 2, two of the optimizations (3 and 4) need not achieve max area or perimeter of overlap.
- Will applying constraints on C1 and C2 (like say, they have same area, same perimeter, both, etc) cause some (or all) of the optimal configs to be the same? Or achieve say, max area of overlap?
- Finaly, what happens if we optimize based not only on the boundaries of C1 and C2 but the entire regions?

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Example 1: When C1 almost reduces to a point and C2 is say a finite square
Minimization (1) will put the small region anywhere on the boundary of square
and (2) will put small region at center of square.

Now, with C1 and C2 interchanged,
Minimization (3) will put small region at center of the square.
Minimization (4) too will have small region at center of square.
So, 3 of the optimizations coincide.
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Example 2: When C1 is a square of side 2 and C2 is a straight line of length 2*sqrt(2):
Minimization (1) - minimizing integral over square of nearest distance to line - will put the line segment as diagonal of square.
Minimization (2) - minimizing integral over square of farthest distance to line - will put line segment as diagonal of square
Minimization (3) - minimizing integral over line of nearest distance to square - will put line as perpendicular bisector of opposite sides of the square.
Minimization (4) - minimizing integral over line of farthest distance to square- will also put line as perpendicular bisector of opposite sides of square.
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Some earlier mathoverflow posts on similar matters:
- https://mathoverflow.net/questions/357124/on-comparing-planar-convex-regions-of-equal-perimeter-and-area
- https://mathoverflow.net/questions/417428/comparing-convex-planar-regions-of-equal-perimeter-and-area-2
- https://mathoverflow.net/questions/409981/on-ways-to-measure-the-difference-between-two-planar-convex-regions

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Note: Instead of minimizing the integral over C1 of distance to farthest poiint on C2 (or with C1 and C2 interchanged), one can think of minimizing the max separation among all pairs of points (P1, P2) where P1 is from C1 and P2 is from C2. This optimization is symmetric between C1 and C2 so there are only 2 questions instead of 4 and in answers to both, one guesses that the centers of Mass of C1 and C2 are coindient.