It is well known that if I have the indicator function $$\iota_S(x)=\cases{ 0 \text{ if } x \in S \\ +\infty \text{ else} }$$ of a convex set $S$, then this is a convex functional and its Fenchel dual is the support function $$\sigma_S(y) = \sup_{x\in S} \langle y,x \rangle .$$
One can then use this in convex optimisations problems to find the dual of the problem itself.
So, if I have $$\min_{x\in C} \psi(x) \text{ s.t.} Ax=b,$$ then I can rewrite the problem as $\min_x \psi(x) + \iota_C(x) + \iota_S(Ax)$, where $\iota_s(x) = \cases {0 \text{ if } x = b,\\ +\infty \text{ else}}$ and compute the dual of my problem, using Fenchel-Rockafeller duality as $\sup_y -\sigma_S(-y) - (\psi+\iota_c)^\star(A^\star y).$
In this simple case, it's easy to see that $-\sigma_S(-y)= \inf_{x\in S} \langle y,x \rangle = \langle y,b \rangle $ (as $x \in S \iff x = b$).
My question then is: what happens if $S$, instead of being $S=\{ x : x = b\}$ is a set denoting inequality constraints? For instance, $S=\{ x: x \geq \gamma\}$? Is there a standard/straight-forward way of characteristing the Fenchel dual?
More precisely, if I have a problem $\min_{x\in C} \psi(x) \text{ s.t } Ax \geq \gamma$ what would the dual be? If our space were the space of Radon Measures and $A$ were a linear operator (say, the integral of the measure with respect to a fixed measurable function $g$) how would the constraint that $\int dQ g \geq \gamma$ impact the dual?
Intuitively, the new $-\sigma_S(-u)$ would be $\inf_{Q\in S} \int u dQ$. If $u=g$ then clearly this infimum can be $\gamma$, otherwise it is $-\infty$. But, how do I reason about all the possible measures and functionals I can plug in this infimum? If $u = \lambda g$ with $\lambda >0$, is it then true that $\inf_{Q\in S}\int u dQ = \inf_{\lambda>0} \lambda \gamma$? Is it arbitrarily small then? What about other (non-linear) transformations of $g$?
Thank you for any suggestion/reference you might provide me!