the Legendre-Fenchel transform
/ luh-ZHAHND FEN-shel /
The Legendre-Fenchel transform is the algebraic operation that converts a log-moment generating function into a rate function, and it is the single piece of convex analysis a large-deviations practitioner uses every day. It is why rate functions in Cramer's and the Gartner-Ellis theorem are convex, and it is the precise dictionary between the 'tilt parameter' world of moment generating functions and the 'deviation value' world of rate functions.
Given a function f on R^d, its Legendre-Fenchel transform (convex conjugate) is f*(x) = sup over theta of (<theta, x> - f(theta)), where <.,.> is the inner product. The supremum picks, for each target x, the tilt theta that best balances the linear reward <theta, x> against the cost f(theta). Geometrically f* records the intercepts of supporting hyperplanes of f. Two facts drive everything: f* is always convex and lower semicontinuous regardless of f, and for a convex lower-semicontinuous f the transform is an involution, (f*)* = f (the Fenchel-Moreau theorem). Differentiability transfers to a dual relation: if f is smooth and strictly convex, then x and theta are conjugate when x = f'(theta), and then f*(x) = theta x - f(theta).
In large deviations one applies this to f(theta) = log E[e^(theta X)], the log-mgf, also called the cumulant generating function. Its convex conjugate is the Cramer rate function I = f*. Because f(0) = 0 and f'(0) = E[X], the conjugate I vanishes precisely at x = E[X] with I(E[X]) = 0, recovering the law of large numbers as the unique zero of the rate function. The convexity of every Cramer/Gartner-Ellis rate function is simply the statement that a convex conjugate is always convex; non-convex rate functions therefore can only arise from non-conjugate constructions such as the contraction principle.
For a standard Gaussian, f(theta) = log E[e^(theta X)] = theta^2/2. Its convex conjugate is f*(x) = sup_theta (theta x - theta^2/2) = x^2/2, attained at theta = x. So the Gaussian rate function x^2/2 is just the self-conjugate parabola — the quadratic is its own Legendre transform up to the trivial scaling.
log-mgf and rate function are convex conjugates of each other.
(f*)* = f requires f to be convex and lower semicontinuous; for a non-convex f the double transform (f*)* returns the convex hull (the largest convex lower-semicontinuous minorant), not f itself. This is exactly why a non-convex landscape cannot be recovered by Legendre duality alone.