Definition 21.4.1E8746DA689A7

Definition 21.4.1.E8746DA689A7 Let $F,G$ be finite convex functions on $N_{\mathbb R}$. As for functions on a compact manifold (section 6.18), we write $F\preceq G$ if $F\leq G+C$ on $N_{\mathbb R}$ for some constant $C$, and $F\sim G$ if $F\preceq G$ and $G\preceq F$. The closed convex set

is nonempty, and by theorem 21.3.2 it is the closure of the set $\partial F(N_{\mathbb R})$ of all supporting slopes of $F$; when it is compact we call it the slope body of $F$. For a convex body $P\subset M_{\mathbb R}$, let $\mathcal P(N_{\mathbb R},P)$ be the set of finite convex functions $F$ on $N_{\mathbb R}$ with $F\preceq h_ P$, and let the bounded class $\mathcal E^\infty (N_{\mathbb R},P)$ be the set of those with $F\sim h_ P$. If $P$ has nonempty interior, the full-mass class $\mathcal E(N_{\mathbb R},P)$ is the set of $F\in \mathcal P(N_{\mathbb R},P)$ with $K_ F=P$; if $P$ has empty interior, we set $\mathcal E(N_{\mathbb R},P)=\mathcal P(N_{\mathbb R},P)$. Finally, the finite-energy class $\mathcal E^1(N_{\mathbb R},P)$ is the set of $F\in \mathcal E(N_{\mathbb R},P)$ with $\int _ PF^*\, \mathrm dp<+\infty $. Since $F^*\geq -F(0)$ by the definition of $F^*$, this integral is well defined in $(-\infty ,+\infty ]$.