As shown above, for a convex function f ( x ) , with x = x ¯ ¯ maximizing or making p x − − f ( x ) bounded at each p to define the Legendre transform f ∗ ∗ ( p ) = p x ¯ ¯ − − f ( x ¯ ¯ ) and with g ≡ ≡ ( f ′ ) − − 1 , the following identities hold.