I am looking for a general method to show that two convex functions have the same minimizer. That is, for a pair of arbitrary convex functions $f$ and $g$, I want to know when it is possible to show that
$x^* = \text{argmin}_x f(x) = \text{argmin}_x g(x)$
Is this possible? Others have asked this kind of question, but they always asked about a specific pair of functions, and the answers they got were fairly specific: see here and here. What I am looking for is a general answer to this question.
I am not sure I understood the generality level you are looking for with your question. I suppose that it is pretty difficult to say something in general, but if you are looking for a simple criterion, then I think immediately at the following:
Let $f$, $g$ two convex real-valued functions. If you can show that $f$ can be obtained from $g$ composing $g$ with a strictly increasing function $h$, then the minimizer of $f = h \circ g$ and $g$ is going to be the same.