A note on sets having the Steinhaus property

A planar set is said to have the Steinhaus property if however it is placed on $R2$, it contains exactly one integer lattice point. The main result proved in this note is that if a set S satisfying the Steinhaus proper- contains a circle of positive radius, then it contains the disk enclosed by the circle. A result of Mihai Ciucu [2] would then imply that any set having the Steinhaus property cannot contain a circle of positive radius in it