Let $K$ be a convex set in a topological vector space $X$. We say $x_0 \in K$ is internal [almost internal] if the collection of $x \in K$ satisfying $$ \inf\{ |c| : x_0 + c(x-x_0) \notin K\} > 0 $$ is equal to $K$ [is dense in $K$].
Can someone give an example of a set which contains a point that is almost internal, but not internal? I imagine there are no examples if $X$ is finite dimensional?