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!



Is minimum of convex envelope the same as minimum of the original function?
If $g(x)\leqslant f(x)-\varepsilon$ for some $x$ in $X$ such that $f(x)$ is the minimum of $f$ on $X$, then the function $h=\max\{g,f(x)\}$ is a convex under-estimator of $f$ such that $h\geqslant g$ everywhere and $h\gt g$ around $x$. This contradicts the definition of $g$.


