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Recent questions showed that roots of a random polynomial tend to lie on the unit circle ("Why do roots of polynomials tend to have absolute value close to 1?"; "Distribution of roots of complex ...

Let $\mathcal{S}$ denote the set of all unit complex-valued $d$-dimensional vectors, i.e., $$ \mathcal{S} \triangleq \left\{ \mathbf{s}\in \mathbb{C}^{d} \mid \mathbf{s}^{\mathrm{H}}\mathbf{s}=1 \...