The generalization to convex functions f : X → → R on a convex set X ⊂ ⊂ R n is straightforward: f ∗ ∗ : X ∗ ∗ → → R has domain X ∗ ∗ = { x ∗ ∗ ∈ ∈ R n : sup x ∈ ∈ X ( ⟨ ⟨ x ∗ ∗ , x ⟩ ⟩ − − f ( x ) ) < ∞ ∞ } and is defined by f ∗ ∗ ( x ∗ ∗ ) = sup x ∈ ∈ X ( ⟨ ⟨ x ∗ ∗ , x ⟩ ⟩ − − f ( x ) ) , x ∗ ∗ ∈ ∈ X ∗ ∗ , where ⟨ ⟨ x ∗ ∗ , x ⟩ ⟩ denotes the dot product of x ∗ ∗ and x .