Legendre Transform

Strictly Convex Function

Let $V$ be an $n$-dimensional vector space, with $e_1,\dots ,e_n$ a basis of $V$ and $v_1,\dots ,v_n$ the associated coordinates. Let $F:V\to \mathbb{R}$, $F=F(v_1,\dots ,v_n)$, be a smooth function. Let $p\in V$, $u={\sum }_{i=1}^n{u}_i{e}_i\in V$. The hessian of $F$ is the quadratic function on $V$ defined by $(d^{2}F{)}_p(u):={\sum }_{i,j}\frac{{\partial }^{2}F}{\partial v_i\partial v_j}(p)u_i{u}_j\text{ .}$ The function F is strictly convex if $(d^{2}F{)}_p\gg 0$, $\forall p\in V$.

Proposition

For a strictly convex function $F$ on $V$, the following are equivalent: (a) $F$ has a critical point, i.e., a point where $d{F}_p=0$; (b) $F$ has a local minimum at some point; (c) $F$ has a unique critical point (global minimum); and (d) $F$ is proper, that is, $F(p)\to +\infty$ as $p\to \infty$ in $V$.

Stable Strictly Convex Function

A strictly convex function $F$ is stable when it satisfies above proposition

Proposition

Let $F$ be any strictly convex function on $V$. Given $\ell \in V^{\ast }$, let $F_{\ell }:V⟶\mathbb{R},F_{\ell }(v)=F(v)-\ell (v)\text{ .}$ Since $(d^{2}F{)}_p=(d^2{F}_{\ell }{)}_p$, $\text{F is strictly convex}⟺\text{Fℓ is strictly convex.}$

Stability Set

The stability set of a strictly convex function $F$ is $S_F=\{l\in V^{\ast }|F_l\text{is stable}\}$

Proposition

The stability set $S_F$ is an open and convex subset of $V^{\ast }$.

Legendre Transform

The Legendre transform associated to $F\in C^{\infty }(V;\mathbb{R})$ is the map $L_F:\begin{array}{rl}[t]V& ⟶V^{\ast }\\ p& ⟼d{F}_p\in T_p^{\ast }V\simeq V^{\ast }\text{ .}\end{array}$

Proposition

Suppose that $F$ is strictly convex. Then $L_F:V\stackrel{\cong }{\to }{S}_F\text{ ,}$ i.e., $L_F$ is a diffeomorphism onto $S_F$.

Dual Function

The dual function $F^{\ast }$ to $F$ is $F^{\ast :}{S}_F\to \mathbb{R},F^{\ast }(l)=-{\min }_{p\in V}{F}_l(p)$

Remark

The reason that we have $-{\min }_{p\in V}{F}_l(p)$ in definition of dual function is to be consistent with the other definition for Legendre transformation for convex functions.

Theorem

$L_F^{-1}:S_F\to V$ is defined as: for $l\in S_F$, the value $L_F^{-1}$ is the unique minimum point $p_l\in V$ of $F_l=F-l$. We have that $L_F^{-1}=L_{F^{\ast }}$