Suppose that $C$ is a closed and bounded convex set which is a subset of euclidean plane and which has the property that for every three points $P,Q,R$ which are on the boundary of $C$ the circle that passes through those three points is a subset of $C$. Is there a non-trivial example of such $C$?
The trivial example would be any disk, because if we choose three points on the boundary of some disk then the circle that passes through those three points is the circle that is the boundary of the disk so it is a subset of the disk.
By intuition it seems that the disks are the only closed and bounded convex sets with this property, but I am not so sure.