f*(p) = sup_x [px - f(x)] — the Legendre transform maps a convex function to its convex dual. The dual encodes the function via its tangent slopes. Involutive: (f*)* = f. Fundamental to thermodynamics, mechanics, and information geometry.
Function
p (slope)0.00
x*(p) = argmax0.00
f(x*)0.00
f*(p) = px*-f(x*)0.00
f*(p) = sup_x [px - f(x)]
Geometric: f*(p) = intercept of support line with slope p
Young's inequality: px ≤ f(x) + f*(p)
Thermodynamics: F = E - TS (T↔S dual) H = E + pV (p↔V dual)