# Is an open midpoint convex set in $\mathbb{R}^n$ always convex?

It can be easily shown that a closed midpoint convex set in $$\mathbb{R}^n$$ is always convex, but it has occurred to me that the counter-examples showing a midpoint convex set may not be convex, are sets that are neither open nor closed.

Can anyone show me an open midpoint convex set in $$\mathbb{R}^n$$ that is not convex?

Let $$U$$ be an open midpoint convex subset of $$\Bbb R^n$$. Let $$P$$, $$Q$$ be distinct points of $$U$$. Let $$L$$ be the line joining $$P$$ and $$Q$$. Then $$L\cap U$$ is an open midpoint convex subset of $$L$$. Thus we can reduce the problem to proving that an open midpoint convex subset of $$\Bbb R$$ is convex.