In the book of Gamelin "Unifrom Algebras" I found the following definition:
Let $K \subset V$ be a convex set in a vector space. An element $x \in K$ is called core point of $K$ if whenever $z \in V$ is such that $x+z \in K$ then also $x-\varepsilon z \in K$ for sufficiently small $\varepsilon >0$.
The preliminary question which I would like to ask is whether someone is familiar with this definition? Because as it is stated I understood it that if $x+z \in K$ then there is $\varepsilon_0 >0$ such that for $\varepsilon_0>\varepsilon>0$ also $x-\varepsilon z \in K$ but apparently (in further proofs) author uses it as it would mean that there is some $\varepsilon >0$ such that $x-\varepsilon z \in K$ so I'm little confused.
I'm also interested in the proof/reference for the proofs of the following properties which are mentioned right after the definition:
-if $K$ is finite dimensional then $K$ is the closure of the set of its core points
-if $V$ is topological vector space and $K$ is such that $int K \neq \emptyset$ then the set of core points of $K$ coincides with $intK$.