Lemma 6.6 . [003Q]

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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 $x_{1},\ldots x_{n}$ .