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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.