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.
Given a positive integer $n$, can every 2D convex region $C$ be divided into $n$ convex pieces, all of the same diameter? The pieces ought to be non-degenerate and have finite area.
If the answer to 1 is yes, how does one minimize the common diameter of the $n$ pieces?
For any $n$, can any $C$ be divided into $n$ convex nondegenerate pieces, all of the same least width?
If 4 has a "yes" answer, how does one maximize the common least width of the $n$ pieces?
These questions have obvious analogs in higher dimensions and other geometries.
Note added on 15th November 2020: As I have just come to know, both question 1 and 3 (existence of partitions into n pieces all of same diameter and into n pieces all of equal least width) have affirmative answers. They follow from the work of Avvakumov, Akopyan and Karasev: Convex fair partitions into an arbitrary number of pieces.
However, the existence proof for $n$ pieces all of same diameter (or same least width) does not directly yield an algorithm to determine a partition with that property.