In mathematics, and particularly convex analysis, the lower convex envelope f ˘ {\displaystyle {\breve {f}}} of a real-valued function f {\displaystyle f} defined on a vector space V {\displaystyle V} is defined pointwise 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 V } . {\displaystyle {\breve {f}}(x)=\sup\{g(x)\mid g{\text{ is convex and }}g\leq f{\text{ over }}V\}.}
The lower convex envelope coincides with the biconjugate f ∗ ∗ {\displaystyle f^{**}} of f {\displaystyle 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 ± ∞ {\displaystyle \pm \infty } . In such cases, one has the inequalities
f ∗ ∗ ( x ) ≤ f ˘ ( x ) ≤ f ( x ) . {\displaystyle f^{**}(x)\leq {\breve {f}}(x)\leq f(x).}
See also Convex hull Lower envelope
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