Corollary 17.3.174DCEFD8869DD

Proof. Convex combinations of potentials with the same singularity type as $\phi $ have that singularity type. For $\mathcal E^1(X,\theta ,\phi )$, convexity follows from the concavity of the extended energy (proposition 17.3.13) and theorem 17.3.14. If $u,v\in \mathcal E(X,\theta ,\phi )$ and $0<t<1$, then $(1-t)u+tv\preceq \phi $, and by multilinearity $\mu _{(1-t)u+tv}$ is a combination of the mixed measures $\langle \theta _ u^ j\wedge \theta _ v^{n-j}\rangle $ with coefficients $\binom nj(1-t)^ jt^{n-j}$. Each of these measures has total mass $m$, as noted at the beginning of this section, so $\int _ X\langle \theta _{(1-t)u+tv}^ n\rangle =m$. The last assertion follows from the monotonicity of the extended energy and theorem 17.3.14. $\square$