We add a bit to Cutting convex regions into equal diameter and equal least width pieces - 2. There, we asked, for example: If we divide a 2D convex region C into n convex pieces such that the maximum diameter among the pieces is a minimum, will we necessarily also get all pieces with same diameter?
Questions:
- If the maximum diameter among n convex pieces into which C is being divided is to be a minimum, will it automatically guarantee a minimum of the average diameter among pieces?
(If minimum diameter among pieces is to be maximized, we have degenerate pieces, although one can also assert that average diameter is then maximized).
One can also ask in the same spirit:
- If the maximum perimeter among n pieces is to be minimized, will it have any automatic implication on the average perimeter among pieces?
- If the maximum (minimum) least width among n pieces from C is to be minimized (maximized), will it have any impact on the average least width (The least width of a region is the least distance between any pair of parallel lines that touch the region)?
Note: These questions can be asked in reverse direction - whether minimizing the average of a quantity has any impact on its maximum value among pieces. There are also 'cross' versions - whether, say, minimizing the max perimeter among pieces has any impact on the max diameter.
