Let $S$ be a closed bounded convex subset of a Banach space $E$.
Definition : A convex subset $F\subseteq S$ is called a face of $S$ if $\forall x\in F,y\in S,z\in S$ such that $x\in]y,z[$ we have $y$ and $z$ in $F$.
I am wondering if this implies that $F$ is closed.
The only approach that I tried that didn't fail (but that I couldn't see through) is the following. For any $x\in S\setminus F$ there exist a continuous linear functional $f_x$ (I can't prove that, it feels like this would use the Hahn Banach theorem and needs $F$ to be closed in the first place) and some $c_x$ such that $f_x(x) < c_x \leq f_x(y)$ for all $y\in F$. I think it is doable to show that $F=\bigcap_{x\in S\setminus F} \{ y: f_x(y)\geq c_x \}$ which would make it closed. I don't think this approach is very promissing but I added it in case it can bring some intuition.
Best Answer
Your guess is wrong in infinite-dimensional Banach spaces. An example is the space of probability mass functions $$\Delta=\{p:\mathbb{N}\to\mathbb{C}\mid\forall n\in\mathbb{N}\colon p(n)\geq 0 \mbox{ and }\|p\|_1=1\}$$ on the natural numbers, which is a closed bounded convex subset of the Banach space $\ell^1=\{x:\mathbb{N}\to\mathbb{C}\mid \|x\|_1<\infty\}$ of absolutely summable sequences endowed with the 1-norm $\|x\|_1=\sum_{n\in\mathbb{N}}|x(n)|$.
The closed faces of $\Delta$ are the spaces $\Delta_I=\{p\in\Delta\colon{\rm supp}(p)\subset I\}$ of probability mass functions supported on subsets $I\subset\mathbb{N}$. But, the space $\Delta$ has many more faces. For each $p\in\Delta$, there is the face $$F(p)=\{q\in\Delta_{{\rm supp}(p)}\colon \sup_{n\in{\rm supp}(p)}q(n)/p(n)<\infty\}.$$ However, $F(p)$ is always dense in $\Delta_{{\rm supp}(p)}$, as it contains the dense set of probability mass functions with finite support in ${\rm supp}(p)$. A reference is Sec. 10 in the preprint Weis, S., A note on faces of convex sets (https://arxiv.org/abs/2404.00832).