The definition of the face of a convex set by a nonnegative affine linear polynomial

Question

My question comes from the paper: https://arxiv.org/abs/0911.2750 (p.2~p.3)

For $n\in \mathbf{N}$

1. Let $X = (X_1,\ldots,X_n)$ be an $n$-tuple of variables.
2. Let $\mathbf{R}[X]$ denote the real polynomial ring in these variables.
3. $\mathbf{R}[X]_d$ denotes its finite-dimensional subspace of polynomials of degree at most $d$.

I have two questions on p.3 of that paper as following:

1. Does the last sentence make the exposed face any different? (For example, if the face is exposed, then the subset $\{x\in S \mid \textit{l}(x) = 0\}$) is not empty.)
2. What about if the face is not exposed?

I do not quite understand the meaning of the green part.

Answer 1

As an example, consider in $\mathbb{R}^2$ the convex hull of the path from $(2,0)$ to $(0,0)$ to $(0,2)$ to $(2,2)$, together with the semicircle $(x-2)^2+(y-1)^2=1$, $x\geq 2$. Then the points $(2,0)$ and $(2,2)$ are nonexposed faces.