PomeTheorem 11.33.4.8810E18F852C The functional $B_\varphi $ is concave on the real space $L^\infty (\mathrm{d}V)$: for $F,G$ in that space,
\[ B_\varphi (tF+(1-t)G) \geq tB_\varphi (F)+(1-t)B_\varphi (G),\qquad 0\leq t\leq 1. \]
Theorem 11.33.48810E18F852C
Proof. Let $u,v\sim \varphi $ solve the equations for $F$ and $G$, and set $H=tF+(1-t)G$ and $D=c_\varphi (F)^ t c_\varphi (G)^{1-t}$. Lemma 13.14.10 and the arithmetic–geometric mean inequality give
\[ \begin{split} \mathrm{MA}^{\mathrm{ext}}_\theta (tu+(1-t)v) & \geq \bigl (t(c_\varphi (F)\mathrm{e}^ F)^{1/n} +(1-t)(c_\varphi (G)\mathrm{e}^ G)^{1/n}\bigr )^ n\, \mathrm{d}V\\ & \geq D \mathrm{e}^ H\, \mathrm{d}V. \end{split} \]
The averaged potential $tu+(1-t)v$ has the ordinary type of $\varphi $. If $c_\varphi (H)<D$, its measure dominates the measure of a solution $w\sim \varphi $ for $H$ by the factor $D/c_\varphi (H)>1$. Applying the singular strict domination principle to $w$ and to every constant translate of the averaged potential, as in the proof of proposition 11.33.2, leads to the same contradiction. Hence $c_\varphi (H)\geq D$, and taking logarithms proves concavity. $\square$
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