The Legendre transform of a finite-valued convex Hamiltonian

Let $n\ge 1$ and let $H:{\mathbb{R}}^n\to \mathbb{R}$ be finite-valued. The Legendre transform (convex conjugate) of $H$ is the function $L:{\mathbb{R}}^n\to \bar{\mathbb{R}}$ (The extended real line $\bar{\mathbb{R}}=\mathbb{R}\cup \{-\mathrm{\infty },+\mathrm{\infty }\}$, its order, and the arithmetic that is left undefined) defined by $L(v):={\sup }_{p\in {\mathbb{R}}^n}(p\cdot v-H(p)),v\in {\mathbb{R}}^n,$ the supremum being taken in $\bar{\mathbb{R}}$ (Every subset of $\bar{\mathbb{R}}$ has a least upper bound and a greatest lower bound in $\bar{\mathbb{R}}$, agreeing with the real supremum and infimum on nonempty sets bounded in $\mathbb{R}$) and the inner product being the Euclidean one (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$). The value $L(v)=+\mathrm{\infty }$ is possible and is not excluded.

In the biconjugate expressions $p\cdot v-(+\mathrm{\infty })$ the convention $p\cdot v-(+\mathrm{\infty }):=-\mathrm{\infty }$ is adopted, so that the supremum defining the biconjugate $L^{\ast }(p):={\sup }_{v\in {\mathbb{R}}^n}(p\cdot v-L(v))$ is an ordinary supremum of a set in $\mathbb{R}\cup \{-\mathrm{\infty }\}$ when the value $+\mathrm{\infty }$ occurs: every $v$ with $L(v)=+\mathrm{\infty }$ contributes the value $-\mathrm{\infty }$ and can be discarded.

When $H$ is convex (Convex and strictly convex functions on Euclidean convex sets), $L$ is the convex conjugate used in the Hopf--Lax construction. For every finite-valued $H$ the transform $L$ is convex and lower semicontinuous, being the pointwise supremum of the affine functions $v\mapsto p\cdot v-H(p)$; no superlinearity, differentiability, strict convexity, coercivity or smoothness of $H$ is assumed at this point. The notation $H^{\ast }$ is used interchangeably with $L$.

• Why the convention is recorded. The supremum over $v\in {\mathbb{R}}^n$ in the biconjugate runs over all of ${\mathbb{R}}^n$ even when $L$ takes the value $+\mathrm{\infty }$; the convention makes each such term $-\mathrm{\infty }$, so those points neither enlarge nor obstruct the supremum, and $L^{\ast }(p)=\sup \{p\cdot v-L(v):v\in {\mathbb{R}}^n,L(v)<\mathrm{\infty }\}$. This is the convention under which the biconjugacy lemma A finite-valued convex Hamiltonian equals its biconjugate is stated, and it is fixed here once for every later use.
• Scope. The transform is defined for every finite-valued $H$; convexity, lower semicontinuity and the affine-supremum representation are consequences of the definition, not hypotheses. The later Hopf--Lax regime adds superlinearity of $H$, which is a separate hypothesis and is what makes $L$ real-valued; no choice principle occurs here.