Definitions: The width of a 2D convex region C is the least distance between any pair of parallel lines that both touch the boundary of C. A width equalizer may be defined as any chord of C that cuts it into 2 pieces of equal width. It is not hard to see that from every point on the boundary of C, at least one width equalizer can be drawn.

Question: What is the relationship between the length of a width equalizer and the widths of the pieces it gives? Will a width equalizer of any given C with maximum possible length always yield 2 pieces of minimum width and a shortest equalizer result in pieces of maximum width?

Guess: a longest width equalizer is parallel to a diameter of C.

Similar questions to above can be asked with reference to diameter instead of width. In 3D, width could be defined as distance between a pair of planes tangential to C and an analogous question would be about the areas of width equalizing planes.