We try to proceed from A claim on the concurrency of area bisectors of planar convex regions
Definitions: Given a planar convex region C, an area bisector of C is any line segment that partitions C into 2 pieces of equal area. A perimeter bisector is any line segment that partitions C into 2 pieces of equal perimeter. Note that both bisectors can also be defined as full lines rather than segments.
- Among all C's of unit diameter, which shape maximizes the difference in orientation between the longest area bisector of C and the longest perimeter bisector of C?
Note: by replacing 'longest' in 1 by 'shortest', we have another question for which I am not sure if the answer is different.
- Which C of unit diameter maximizes the ratio between lengths of the longest (shortest) area bisector and the longest (shortest) perimeter bisector?
Guess: triangles seem good candidates as answers for all above questions but I have no quantitative answers.