Lower convex envelope

In mathematics, and particularly convex analysis, the lower convex envelope ${\displaystyle {\breve {f}}}$ of a real-valued function ${\displaystyle f}$ defined on a vector space ${\displaystyle V}$ is defined pointwise as the supremum of all convex functions that lie under that function, i.e.^[1]

$${\displaystyle {\breve {f}}(x)=\sup\{g(x)\mid g{\text{ is convex and }}g\leq f{\text{ over }}V\}.}$$

The lower convex envelope coincides with the biconjugate ${\displaystyle f^{**}}$ of ${\displaystyle f}$, but can differ with the biconjugate when this definition is extended to functions whose domains are not the entire vector space or whose values are extended reals, i.e., allowing ${\displaystyle \pm \infty }$. In such cases, one has the inequalities^[1] $${\displaystyle f^{**}(x)\leq {\breve {f}}(x)\leq f(x).}$$