Example 1

Every proper closed convex cone in $E$, a vector space of finite dimension, has a compact sole. In fact, by II, p. 52, prop. 11 we need only consider the case where $E = \mathbf{R}^n$ and $C = \mathbf{R}_+^n$. If $(e_i)_{1 \leq i \leq n}$ is the canonical basis of $\mathbf{R}^n$, it is clear that the compact convex set which is the convex envelope of the $e_i$ ($1 \leq i \leq n$) is a compact sole for $\mathbf{R}_+^n$.