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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 $\text{ci}(A)=A$.

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  • $\begingroup$ I'm not certain what your question is. It seems that if you write down a condition for "interior" you will have answered your own question. But if you have no criterion for interior, your question can not be answered. $\endgroup$
    – Ryan Budney
    Commented May 5, 2021 at 3:59
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    $\begingroup$ I am mainly asking if this definition is well known or equivalent to some well-known definitions. $\endgroup$
    – Lemma1
    Commented May 5, 2021 at 4:33
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    $\begingroup$ I see the following cases: 1. A is an affine subspace, in which case there are no convex interior points. 2. A is a pointed cone in which case the cusp is the only ci point. 3. All other cases, in which every point is ci. $\endgroup$
    – user130903
    Commented May 5, 2021 at 5:37
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    $\begingroup$ The weird thing about this definition is that it is non-local. As the cone shows, if I know $A$ only in a neighborhood of $a$, then there are cases where I cannot decide if $a\in ci(A)$ or not. Possibly a more natural condition would be "For all $b\notin A$, there exists $\epsilon$", but I think that should be equivalent to the topological interior. Maybe the important question is, what are you trying to achieve with this definition? $\endgroup$
    – mlk
    Commented May 5, 2021 at 8:11
  • $\begingroup$ In Köthe's book Topological Vector Spaces there is a notion of internal points of a convex set $A$ which means that $x$ is an interior point of the intersection of $A$ with every real line through $x$ (of course, interior refers to the usual topology of the reals). $\endgroup$
    – Jochen Wengenroth
    Commented May 5, 2021 at 14:56

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