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For the unit sphere $\\mathbb{S}^d \\subset \\mathbb{R}^{d+1}$, we obtain the precise asymptotic that $\\mathbb{E}\\rho(X_N)[N/\\log N]^{1/d}$ has limit $[(d+1)\\upsilon_{d+1}/\\upsilon_d]^{1/d}$ as $N \\to \\infty $, where $\\upsilon_d$ is the volume of the $d$-dimensional unit ball. 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