We add a little bit to this post: https://mathoverflow.net/questions/356279/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.