I am struggling with proving or disproving the following:
Let $X$ be a locally convex space and let $C\subset X$ be closed convex with non-empty algebraic interior which we denote by $C^i$ (recall that $x\in C^i$ iff $\forall v\in X$, there is $t>0$ such that $x+tv\in C$). Now, is the following statement (S) true:
$(S):\ \forall x\in C^i$, there is a neighborhood $V$ of $x$ such that $V\cap C\subset C^i$.
In other words is $C^i$ (topologically) relatively open in $C$?
Personally I think that it is false, but I could not find a counter-example.