Does every ball of a boundary point of a convex set contain interior points?

Question

Let $(X,d)$ be a metric space and $C\subseteq X$ be a convex set with non-empty interior and non-empty boundary. Furthermore define $$B_{\varepsilon}(x)\triangleq\left\{y\in X\ \big\rvert\ d(x,y)<\varepsilon\right\}.$$ Let now $z\in C$ be a point on the boundary. I suspect that $$B_{\varepsilon}(z)\cap\operatorname{int}(C)\ne\emptyset,$$ for $\varepsilon >0$. Is this true? If so, how could this be proven?

Answer 1

It is true. Hint: show that if $z \in C$, $B_r(x) \subseteq C$, and $\lambda \in (0, 1]$, then $$B_{\lambda r}(\lambda x + (1 - \lambda) z) \subseteq C.$$