Lemma 6.6 (003Q) · ScalingStacks

Lemma 6.6. [52, Lemma 4.3] The fibrewise $T^{n}$ average function

\bar{\phi}(x_{1},\ldots x_{n})=\frac{1}{(2\pi)^{n}}\int_{T^{n}}\phi(e^{x_{1}\log|t|+i\theta_{1}},\ldots e^{x_{n}\log|t|+i\theta_{n}})d\theta_{1}\ldots d\theta_{n}

is a convex function in the variables $x_{1},\ldots x_{n}$ .

003R

Proof. Since the function $\bar{\phi}$ is an average of psh functions, it is psh as a $T^{n}$ -invariant function on $\text{Log}_{t}^{-1}(U)$ . Such functions correspond to convex functions downstairs. ∎