Lower convex envelope

Short description: Mathematics concept

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.

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

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 $\pm \mathrm{\infty }$. In such cases, one has the inequalities $$f^{**}(x)\le \breve{f}(x)\le f(x).$$

See also

• Convex hull
• Lower envelope