Let's say we are given a function $f:\mathbb R ^d\to \mathbb R$ continuous. Assume that $\mathcal F$ is a convex cone of continuous functions ($\mathbb R^d$ to $\mathbb R$) closed under maxima. I am interested in learning more about the $\mathcal F$-envelope of $f$. Let us denote it by $\text{env}_\mathcal F f$, that is, the greatest function in $\mathcal F$ dominated by $f$.
For motivation, take $\mathcal F$ to be the set of convex functions. Then we recover the classical convex envelope, which in particular can be written as $$\text{conv}(f)(y)=\inf_{\mu,\; \text{mean}(\mu)=y} \int f d\mu.$$
Is there something like this for general convex cones $\mathcal F$? Where can I read up on this stuff? I am open to other extra assumptions, as I am still not exactly sure what I'm after.
Many thanks for any help!
