A set X is convex if $\forall x,y \in X$, $\frac{1}{2}(x+y)\in X$

Question

Prove or disprove the following statement: A set X is convex if $\forall x,y \in X$, $\frac{1}{2}(x+y)\in X$.

A set X is convex is $\forall x,y\in X, \lambda \in [0,1] \implies \lambda x + (1-\lambda)y \in X$.
I think that the statement doesn't hold and I tried to disprove it formally, but I couldn't. Should I be looking for counterexamples? Any hints?

Answer 1

That looks very much like a condition that $\Bbb Q$ would satisfy.