Let $f: X\to \mathbb{R}$ be bounded. Consider the following definitions:
Definition 1: $f^{*}(x)=\inf\{h(x): h\in \mathbb{R}^{X} \;\;\text{affine and}\;\; h\ge f\}$.
Definition 2: $f^{**}(x)=\inf\{h(x): h\in \mathbb{R}^{X} \;\;\text{affine, continuous and}\;\; h\ge f\}$.
Definition 3: $f^{***}(x)=\min\{h(x): h\in \mathbb{R}^{X} \;\;\text{concave and}\;\; h\ge f\}$.
All were extracted from different textbooks or articles. They refer to the concave envelope of $f$.
Are they all equivalent?
I can see how they are related, but, for instance, for $g$ strictly concave with $g(x)\ge f^{***}(x)$, $f^{***}(x)=g(x)$ but what about $f^{*}(x)$? Why is continuity required in definition 2?
Posed as it is posed, for general $X$ (hence possibly infinite dimensional) it is clear that there may exist linear unbounded functions, making the two definitions nonequivalent.
Note also that for general $X$ we may not even be sure the $\min$ exists in definition $3$.
Knowing what is $X$ (a generic set, a set endowed with a preference relation and maybe an order topology, a finite dimensional space,...) is fundamental for the question, since in total generality it is false.