Counterexample of convex property

Question

I have studied a property in my textbook which establishes that, given $\emptyset \neq X \subset \mathbb{R} ^ n $ and $\alpha \in [0,1]$,$X \subset (1-\alpha)X+\alpha X$, whose proof is evident since the set $$(1-\alpha)X+\alpha X=\{y \in \mathbb{R} ^ n| y=(1-\alpha) x+\alpha x, x\in X\} $$ and then $$x_0 \in X \Rightarrow (1-\alpha)x_0+\alpha x_0=x_0$$. In addition, the double inclusion it is guaranteed when $X$ is a convex set. But in general it is not guaranteed and I am trying to find a counterexample for it. My intuition says a sphere may work but I can't see really how to formalize it. Aprreciate any help. Thanks.