Let $\lambda \notin [0, 1]$ and $x \neq y$ such that $x,y\in \partial C$. Then $\lambda x + (1-\lambda)y \notin \overline C$

Question

Let $X$ be a topological vector space, $C$ a convex subset of $X$, and $x,y\in \partial C$. Suppose $\lambda \notin [0, 1]$ and $x \neq y$. I would like to ask if $z:= \lambda x + (1-\lambda)y \notin \overline C$. This is intuitively true when $C$ is a line segment or a circle in $\mathbb R^2$. However, I do not have a proof.

Could you elaborate on this conjecture?

Answer 1

I found that the conjecture is not correct. A counter example is a square in $\mathbb R^2$.