Hello everyone my question is:

$Question:$ Consider a function $f:X \rightarrow \mathbf R$ where $X$ is a convex subset of $\mathbf{R}^n$. The convex envelope of $f$ over $X$ is defined as the pointwise supremum of convex under-estimator of $f$, denoted as $g$. I want to ask if the minimum of $f$ over $X$ is the same as the minimum of $g$ over $X$?

This sounds intuitive but I am not sure if this is true. If yes, how to prove it rigorously? And then does it mean we can use convex envelopes to replace any non-convex function to make some hard non-convex optimization easy?

Thank you in advance!