**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.

1. 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/perimeters* of width equalizing *planes*. 

2. Further to https://mathoverflow.net/questions/449851/a-claim-on-concurrency-of-width-bisectors-of-planar-convex-regions, does the concurrency of all width equalizers guarantee that C is centrally symmetric? What are the implications of every point on the boundary having a unique width equalizer starting at that point?