Convex set equals convex functions within optimization?

Question

Can optimizing a convex function subject to convex constraints be written as optimizing the function subject to a convex set? Does the intersection of convex nonlinear ineualities necessarily describe a convex set? In notation, is minimize $f(x)$ subject to $g_i(x)\le0, \forall i$, where $f$ and $g_i$ are convex functions equivalent to minimizing $f$ over a convex set $\mathcal{C}$ corresponding to the constraints (if that is true)?

Answer 1

We know that a convex function is also quasiconvex and therefore has lower level sets that are convex sets. Thus if $g_i$ is convex then the set of $x$ such that $g_i(x)\leq 0$ is a convex set. The set $\mathcal{C}$ is the intersection of these sets over $i$. Since the intersection of convex sets is convex, $\mathcal{C}$ is convex.

See https://en.wikipedia.org/wiki/Convex_optimization#Convex_optimization_problem