ScalingStacks

Proof. The if direction is because the asymptotic growth condition (18) implies the gradient of $u$ must be contained in $\Delta$.

For the only if direction, we apply the Legendre transform:

$$u^{*}(p)=\sup_{x\in\partial\Delta_{\lambda}^{\vee}}\{\langle x,p\rangle-u(x)\},\quad p\in\Delta,$$

and consider a version of the double Legendre transform

$$u^{**}(x)=\sup_{p\in\Delta}\{\langle x,p\rangle-u^{*}(p)\}.$$

Clearly $u^{**}$ is convex, and admissible by the boundedness of $u^{*}$, and $u^{**}(x)\leq u(x)$ on $\partial\Delta_{\lambda}^{\vee}$ because

$$\langle x,p\rangle-u^{*}(p)\leq u(x),\quad\forall p\in\Delta.$$

Our characterisation precisely ensures $u^{**}(x)\geq u(x)$ on $\partial\Delta_{\lambda}^{\vee}$. Then $u^{**}$ provides the canonical extension. ∎