Let I ⊂ ⊂ R be an interval, and f : I → → R a convex function ; then the Legendre transform of f is the function f ∗ ∗ : I ∗ ∗ → → R defined by f ∗ ∗ ( x ∗ ∗ ) = sup x ∈ ∈ I ( x ∗ ∗ x − − f ( x ) ) , I ∗ ∗ = { x ∗ ∗ ∈ ∈ R : f ∗ ∗ ( x ∗ ∗ ) < ∞ ∞ } where sup denotes the supremum over I , e.g., x in I is chosen such that x ∗ ∗ x − − f ( x ) is maximized at each x ∗ ∗ , or x ∗ ∗ is such that x ∗ ∗ x − − f ( x ) as a bounded value throughout x exists (e.g., when f ( x ) is a linear function).