This post is a spinoff from Cutting convex regions into equal diameter and equal least width pieces

**Definitions:** The diameter of a convex region is the greatest distance between any pair of points in the region. The least width of a 2D convex region can be defined as the least distance between any pair of parallel lines that touch the region.

- Consider dividing a 2D convex region C into n convex pieces such that the maximum diameter among the pieces is a minimum. Will such a partition necessarily require all pieces to have the same diameter? This looks unlikely but I have no counter example.

**Remark:** Maximizing the least diameter among n convex pieces can be seen to have no neat solution - with most of the pieces near-degenerate, one can achieve, for each piece a diameter arbitrarily close to the diameter of C itself. However, I can't think of any quantity such that if we try to *minimize the maximum* of this quantity among n convex pieces, we would get degenerate pieces.

- If the lowest least width among n convex pieces into which C is being cut ought to be maximized, will such a partition necessarily be one where all pieces have same least width? Again, one has no counter example.

Note 1: For both questions, one might have a "not true in general but true for sufficiently large and finite n" answer. But this is a guess.

Note 2: Not sure if question 2 is related to the Plank Problem. Maybe not because maximizing the lowest least width of the pieces appears to favor triangular pieces rather than planks.

Note 3: From question 2, one can derive what seems to be a bunch of related questions: Given a positive integer n, find the smallest convex region C ("smallest" could mean least area, least diameter or least perimeter) such that from C, n convex regions can be cut with the least width of each being at least equal to unity.

**Further Thoughts:** If maximum (minimum) area among n convex pieces is to be minimized (maximized), then, it is easy to see all pieces should have same area. Maximizing (minimizing) the minimum (maximum) *perimeter* among n convex pieces also probably equalizes the perimeter among pieces(I have no proof) - for the n=2, this is easily true.

Note (13th November 2021): Another quantity that can be considered for minimizing the maximum (maximizing the maximum) among n pieces is the moment of inertia.

*A guess:* To maximize the least perimeter among n convex pieces cut from a convex region C, at least one of the cut lines necessarily ends at an end of a diameter of C.