The sets are defined in $\mathbb{R}_+^n$ $(n\geq 1)$. The relative interior of a convex set $C$ is denoted $\mathring C$.

Let $S$ be the $n$-simplex:

$$S=\left\{x\in\mathbb{R}_+^n,\,\sum_{i=1}^n x_i=1\right\}$$

and $E$ be a linear subspace such that $E\cap\mathring S\neq\varnothing$ (hence, $E$ contains a vector with positive coordinates) of codimension $2$ in $\mathbb{R}^n$. For $F$ a face of $E\cap S$, one can see that there exists a unique face $G$ of $S$ such that $\mathring F\subset\mathring G$. Hence, $E\cap G=F$.

Question: We assume that, in $\mathring G$, $\mathring F$ has codimension $2$. If a face $G'$ of $S$ satisfies $G'\cap E=F$, do $G'=G$?

The sets are defined in $\mathbb{R}_+^n$ $(n\geq 1)$. The relative interior of a convex set $C$ is denoted $\mathring C$.

Let $S$ be the $n$-simplex:

$$S=\left\{x\in\mathbb{R}_+^n,\,\sum_{i=1}^n x_i=1\right\}$$

and $E$ be a linear subspace such that $E\cap\mathring S\neq\varnothing$ (hence, $E$ contains a vector with positive coordinates) of codimension $2$ in $\mathbb{R}^n$. For $F$ a face of $E\cap S$, one can see that there exists a unique face $G$ of $S$ such that $\mathring F\subset\mathring G$. Hence, $E\cap G=F$.

Question: We assume that, in $\text{Aff}(G)$, $\text{Aff}(F)$ has codimension $2$. If a face $G'$ of $S$ satisfies $G'\cap E=F$, do $G'=G$?