Consider any compact metric space $S$ which is a convex subset of a vector space. I am trying to see if the following definition makes sense, or is equivalent/close to some well-known definitions.

Consider a convex subset $A\subsetneq S$. Let the "convex interior" of $A$ (I made the name up), denoted as $\text{ci}(A)$, be defined as follows: $$ \text{ci}(A)=\{a\in A| \exists \epsilon\in(0,1) \text{ and } b\not\in A \text{ such that } \epsilon a+(1-\epsilon) b \in A\}. $$

It is not hard to see that $\text{int}(A)\subseteq \text{ci}(A)$. But in certain cases $\text{ci}(A)$ is a larger set. For example, let $S$ be the unit square in $\mathbb{R}^2$ and $A=\{(x,y)\in S| 0.1\le x\le 0.2\}$. Here $\text{ci}(A)=A$.

Consider any compact metric space $S$ which is a convex subset of a vector space. I am trying to see if the following definition is equivalent/close to some well-known definitions.

Consider a convex subset $A\subsetneq S$. Let the "convex interior" of $A$ (I made the name up), denoted as $\text{ci}(A)$, be defined as follows: $$ \text{ci}(A)=\{a\in A| \exists \epsilon\in(0,1) \text{ and } b\not\in A \text{ such that } \epsilon a+(1-\epsilon) b \in A\}. $$

It is not hard to see that $\text{int}(A)\subseteq \text{ci}(A)$. But in certain cases $\text{ci}(A)$ is a larger set. For example, let $S$ be the unit square in $\mathbb{R}^2$ and $A=\{(x,y)\in S| 0.1\le x\le 0.2\}$. Here $\text{ci}(A)=A$.