region between two concentric circles are not convex set in eucledian space of order 2. I want a counterexample to show that the region between two concentric circles in $\mathbb{R}^2$ is not a convex set. I think we need to find two points in the common region of concentric circles and then will show that their convex linear combination is not in $\mathbb{R}^2$, but I am unsure of this.

The counteraxample is straightforward: Any straight line through the common center passes (in this order) through big circle, small circle, centre, small circle, big circle. This makes the center a convex combinatoin of points in the first and the second annulus segment encountered.