We add a little bit to this post: On 'fair bisectors' of planar convex regions.

Definitions: Given a planar convex region C (could be smooth or polygonal), an area bisector of C is any line that partitions C into 2 pieces of equal area. A perimeter bisector is a line that partitions C into 2 pieces of equal perimeter. Obviously, thru every point on the boundary of C we can draw an area bisector and a perimeter bisector.

Question: Are the following claims easy to prove/counter?

• A planar convex region is centrally symmetric if and only if its area bisectors are all concurrent.
• A planar convex region is centrally symmetric if and only if its perimeter bisectors are all concurrent.

Note: Higher dimensional analogs of these claims are easy to state.