Lower convex envelope

In mathematics, the lower convex envelope ${\breve {f}}$ of a function $f$ defined on an interval $[a,b]$ is defined at each point of the interval as the supremum of all convex functions that lie under that function, i.e.

$f(x)=\sup\{g(x)\mid g{\text{ convex and }}g\leq f{\text{ over }}[a,b]\}.$