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.
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![{\displaystyle [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935)
![{\displaystyle {\breve {f}}(x)=\sup\{g(x)\mid g{\text{ is convex and }}g\leq f{\text{ over }}[a,b]\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d1f960800ad9743d30c99408b53d9adc0b88eb1)
