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

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