We try to proceed from https://mathoverflow.net/questions/448712/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.

1. 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.

2. 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.