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.
Now let $$U\subseteq\Bbb R$$ be open and midpoint convex. Let $$P$$, $$Q\in U$$ be distinct. We can assume $$P=0$$, $$Q=1$$. There is $$\newcommand{\ep}{\varepsilon}\newcommand{\sub}{\subseteq}\ep>0$$ such that $$(-\ep,\ep)\sub U$$ and $$(1-\ep,1+\ep)\sub U$$. By midpoint convexity, $$(1/2-\ep,1/2+\ep)\sub U$$, $$(1/4-\ep,1/4+\ep)\sub U$$, $$(3/4-\ep,3/4+\ep)\sub U$$, etc. Thus $$(r-\ep,r+\ep)\sub U$$ for all dyadic rationals $$r\in[0,1]$$. But these intervals cover $$[0,1]$$. Then $$[0,1]\sub U$$ and $$U$$ is convex.