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 $(x-2)^2+(y-1)^2=1$, $x\geq 0$. Then the points $(2,0)$ and
$(2,2)$ are nonexposed faces.