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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.

  1. Among all C's of unit diameter, which shape maximizes the difference in orientation between a longest area bisector of C and that longest perimeter bisector of C that is closest in orientation to it?

Note 1: Thanks to Daniel Asimov for pointing out that there could be multiple longest (and shortest) bisectors of both kinds.

Note 2: by replacing 'longest' in 1 by 'shortest', we have another question for which I am not sure if the answer is different.

  1. Which C of unit diameter maximizes the ratio between lengths of a longest (shortest) area bisector and a longest (shortest) perimeter bisector?

Guess: triangles seem good candidates as answers for all above questions but I have no quantitative answers.

  1. Are there C's for which there is only one longest (shortest) area bisector and many longest (shortest) perimeter bisectors (or vice versa)? In general, given arbitrary integers m and n, can one always construct a C with m longest (shortest) area bisectors and n longest (shortest) perimeter bisectors? This question follows from the comment below from Daniel Asimov.
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    $\begingroup$ What does "the longest" mean in case there are more than one longest bisectors (of area or perimeter)? $\endgroup$
    – Daniel Asimov
    Commented May 2 at 16:27
  • $\begingroup$ Thank you. Made an edit and also added a subquestion that seems to follow from your comment. $\endgroup$
    – Nandakumar R
    Commented May 2 at 17:07
  • $\begingroup$ I suggest eliminating all instances of the phrase "the longest" and replacing each with "a longest". $\endgroup$
    – Daniel Asimov
    Commented May 2 at 17:19

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