Remark 2

Suppose further that $X$ is a closed convex cone with vertex 0 and that $f$ is positively homogeneous. Then $f$ is the upper envelope of a family of functions which are the restrictions to $X$ of continuous linear forms in $E$. For, let $(u_\alpha)$ be a family of continuous affine linear functions in $E$ of which the restrictions to $X$ have $f$ as their upper envelope. Put $u_\alpha = v_\alpha + \lambda_\alpha$, where $\lambda_\alpha \in \mathbf{R}$, and where $v_\alpha$ is a continuous linear form in $E$. We have $\lambda_\alpha = u_\alpha(0) \leq f(0) = 0$. On the other hand, if $x \in X$, we have for every $\mu > 0$,

therefore $u_\alpha \leq v_\alpha \leq f$ in $X$ so that $f$ is the upper envelope of the $v_\alpha$.