PomeDefinition 21.11.41.2AEBD4E3830A We define the set $\mathcal{P}(N,P)$ as the set of proper convex functions $F\in \mathrm{Conv}(N)$ such that $F\preceq \operatorname{Supp}_ P$.
We define the set $\mathcal{E}^{\infty }(N,P)$ as the set of closed convex functions $F\in \mathrm{Conv}(N)$ such that $F\sim \operatorname{Supp}_ P$.
We define the set $\mathcal{E}(N,P)$ as follows: Suppose that $\operatorname{Int}P=\varnothing $. Then $\mathcal{E}(N,P)\coloneqq \mathcal{P}(N,P)$; otherwise, let
\[ \mathcal{E}(N,P)=\left\{ F\in \mathcal{P}(N,P) : P=\overline{\nabla F(N)} \right\} . \]
We define the set $\mathcal{E}^1(N,P)$ as the subset of $\mathcal{E}(N,P)$ consisting of $F\in \mathcal{E}(N,P)$ with
\[ \int _ P F^*\, \mathrm{d}\operatorname{vol}<\infty , \]
where $\mathrm{d}\operatorname{vol}$ is any Lebesgue measure on $M$.
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