We add a little to On 'fair bisectors' of planar convex regions and A claim on the concurrency of area bisectors of planar convex regions .
A fair bisector of a planar convex region is a line that cuts the region into 2 convex pieces of equal area and equal perimeter.
Evidently, if a planar convex region is centrally symmetric, for every boundary point, the line thru it and the center will be a fair bisector. Here we ask about the converse:
- If a planar convex region is such that there is a fair bisector passing through every one of its boundary points (no further conditions specified), is the region necessarily centrally symmetric?
