In mathematics, the lower convex envelope f ˘ {\displaystyle {\breve {f}}} of a function f {\displaystyle f} defined on an interval [ a , b ] {\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.

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