Is this convex set open? [closed]

Let $X$ be a normed space over $\mathbb{R}$, and let $K \subset X$ be a convex subset with the property that, for every $u \in X, u \neq 0$ there exists $M(u) > 0$ such that

$ \{ \lambda \in \mathbb{R} : \lambda u \in K \} = (-M(u), M(u))$.

Is this set open? My intuition tells me that it must be, but I can't find a way to show it.