How to prove the condition for interior points in convex sets?

Question

$C$ is a non-empty convex set in $\mathbb{R}^n$. If $\forall y \in \mathbb{R}^n,$ there exists some $\epsilon>0$ such that $z+\epsilon y\in C$, then $z\in \rm{interior}(C)$

I cannot prove this conclusion. How to prove this theorem?

Answer 1

I assume you consider the standard metric topology in $\mathbb{R}^n$, induced by the Euclidean norm. Let $y=e_i$, $i=1,2,\dots n$, $e_i=(0,0,0,\dots 1,\dots 0)$ where the $1$ is in the $i$-th position. For each one of these you have some $\epsilon_i>0$ satisfying the given condition. Take $\epsilon=\min \epsilon_i$. By convexity, the set $z+\textrm{conv}\, \{0,\pm \epsilon e_{i_1}, \pm \epsilon e_{i_2},\dots \pm \epsilon e_{i_n}\}$ with $i_k\neq i_j$ for $k\neq j$ is contained in $C$. But the ball centered at $z$ with radius $\epsilon/\sqrt{n}$ is contained in the above set (think of the two or three dimensional cases). Therefore that ball is contained in $C$ and $z$ is interior.