Let $X \subset \mathbb{R}^n$ be convex, as well as the functions $f:X \to \mathbb{R}$ and $g:X \to \mathbb{R}$.
Prove $h(b)=\inf \{f(x): g(x) \le b, x \in X \}$ defined on $\mathbb{R}$ is convex also.
I've tried the supporting hyperplane theorem but similarly i don't think that's the way to go.
Define $I(g(x)\leq y)$ as the convex indicator function (taking value $0$ if the constraint is satisfied, $\infty$ otherwise), and define $k(x,y) = f(x) + I(g(x)\leq y)$. Since $k$ is jointly convex and $X$ is convex, $h(y) = \inf_{x\in X} k(x,y)$. Since $k$ is a convex function. This result is known as "partial minimization of a jointly convex function preserves convexity".
The proof of "partial minimization preserves convexity" typically goes via epigraphs. The epigraph of $k$ is $\{(x,y,z) : f(x) \leq z, g(x)\leq y, x \in X\}$, which is a convex set. The projection of a convex set onto some of its components is convex: $\{(y,z) : \exists x \in X : f(x) \leq z, g(x)\leq y\}$. This is exactly the epigraph of $h$, so $h$ is also convex.