Lower convex envelope

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

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

The lower convex envelope coincides with the biconjugate $f^{**}$ of $f$, 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 $\pm \infty$. In such cases, one has the inequalities $$f^{**}(x)\leq {\breve {f}}(x)\leq f(x).$$

See also

• Convex hull
• Lower envelope