Let $X$ be an infinite-dimensional real vector space, and $C\subseteq X$ a non-empty convex cone. Here are some standard definitions. $C$ is algebraically closed if and only if, for all $x\neq y\in X$, $[x,y)\subseteq C\implies y\in C$. And $c\in C$ is in the intrinsic core of $C$ if and only if, for any $x\in C\setminus\{c\}$, there exists $y\in C$ with $c\in(x,y)$.
In general, $C$ may have an empty intrinsic core. But what if $C$ is algebraically closed?
Here is an example of $C$ algebraically closed with an empty intrinsic core.
Let $X$ be the space of all sequences of real numbers with finitely many non-zero terms, and $C$ the cone of such sequences with all terms non-negative.