Let $(X,\tau)$ be a topological vector space, $D \subset X$ and $(D, \tau_{D})$ a topological space. We are working on this subspace topology.
Suppose we have a closed convex set $C \subset D$, which has a non-empty interior, $C^{\mathrm{o}}$.
If $x \in \partial C$ and $y \in C^{\mathrm{o}}$, then for any $\lambda \in (0,1)$, is $\lambda x + (1-\lambda)y \in C^{\mathrm{o}}$?
I think this is true, but I am not sure how to prove it. Both answers to this (especially the answer which is not accepted) Closure of interior of closed convex set may be helpful.
The point $y$ is in the interior of $C$, hence for sufficiently small $\varepsilon>0$, the $\varepsilon$-ball around $y$ is also in $C$. As $C$ is convex, the line from any point in the $\varepsilon$-ball to $x$ is also in $C$. The union of all these lines forms a kind of cone that is part of $C$. For any point on the line from $x$ to $y$ that is not $x$ itself a small ball around this point is inside this cone and thus in $C$. This proves that any point on this line apart from $x$ is not on the boundary of $C$.