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Is there a closed-form expression for the integral of max($x_1,..., x_n$) over the ($n-2$)-dimensional hypersphere, {$x \in \mathbf{R}^n$: $\sum_{i=1}^n x_i = 0$, $\sum_{i=1}^n x_i^2 = 1$}?

I come across this integral in trying to find the expected value of the maximum z score from independent normals.

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