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Following A claim on the concurrency of area bisectors of planar convex regions, let me record a couple of simple queries.

An area bisector (perimeter bisector) of a planar convex region is a chord that cuts the region into two equal area (equal perimeter) pieces.

  1. If all area bisectors of a planar convex region are of same length (no other condition specified), is the region a circular disk?

  2. If all perimeter bisectors of a planar convex region have same length, is the region a circular disk?

Note 1: If the answer to both above is "not necessarily", then one can ask if all area bisectors being equal length and all perimeter bisectors being equal length is enough to make the region a circular disk.

Note 2: analogous questions can be asked with area/perimeter replaced by other quantities such as moment of inertia, width and so forth.

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This paper has a reference to a positive answer to the first question.

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  • $\begingroup$ Thank you. Indeed, there appear to be infinitely many zindler curves with convex interior that answer both above questions. As quoted, he has proved that “a convex set has constant area halving distance iff it has constant perimeter halving distance” $\endgroup$
    – Nandakumar R
    Commented Jun 29 at 18:31

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