Numerator: d_m(P_m, Q_m)^p

The passage around this formula

Beyer and colleagues proved the canonical result. Under broad conditions on the data and query distributions — much broader than independence and identical distribution across dimensions — as dimensionality rises the distance to the nearest data point approaches the distance to the farthest. Formally, writing $d_m$ for the distance function in m dimensions, $P_m$ for a data point and $Q_m$ for a query point: $\lim_{m \to \infty} \operatorname{var}\!\left( \frac{d_m(P_m, Q_m)^p}{\mathbb{E}\left[ d_m(P_m, Q_m)^p \right]} \right) = 0 \;\;\Longrightarrow\;\; \lim_{m \to \infty} \Pr\!\left[ D_{\max}^{(m)} \le (1 + \varepsilon)\, D_{\min}^{(m)} \right] = 1$. for every $\varepsilon$ > 0 [ 11 ] . The condition is on the relative variance of the distance distribution: when distances stop varying much relative to their own mean, the nearest neighbour stops being distinguishable from everything else. The authors call such a query…

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