Convex partitions - averages of quantities

We continue from the following posts:
- https://nandacumar.blogspot.com/2022/03/max-of-min-and-min-of-max-3.html
- https://nandacumar.blogspot.com/2021/04/convex-partitions-max-of-min-and-min-of.html
and this post at mathoverflow:
- https://mathoverflow.net/questions/376672/cutting-convex-regions-into-equal-diameter-and-equal-least-width-pieces-2
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Some background:

If a convex planar region C is cut into n convex pieces such that the average perimeter of pieces is maximized, will all pieces necessarily have same perimeter?

The answer appears to be "not necessarily". Consider a disk being cut into 3 pieces such that the sum of perimeters is maximized; the best seems to be two half disks and one infinitely thin piece along the diameter.

What if we try to maximize the geometric mean rather than arithmetic mean of all perimeters?

Remark 1: If we minimize the average perimeter, n-1 of the pieces will shrink to points and one piece will be C itself - not too interesting. If instead of perimeter, we try to maximize the average diameter, all pieces will try to contain a diameter of C and most will become degenerate.

Remark 2: Any partition whatever of C into n pieces will give the same arithmetic mean for *area* of pieces. Howeever if the geometric mean of area is to be maximized, then areas ought to be equal.
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Some possibly harder and more interesting questions:
Definition: 'width' of a convex region is the distance between the pair of parallel lines that touch it and are least apart.

1. If we try to maximize the average over pieces of the least width, will all pieces necessarily have equal least width? Again, average could be taken to mean arithmetic or geometric mean.
2. What if we consider the average width of each piece?

3. After perimeter and width, one can think of 'centralness' - maximizing the average of centralness. For definition, see https://mathoverflow.net/questions/396579/a-center-of-convex-planar-regions-based-on-chords