Definitions: The least width of a 2D convex region C is the least distance between any pair of parallel lines that both touch the boundary of C (in what follows, we refer to this quantity as simply 'width'). 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.
- 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.
- Further to 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?
