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Aggarwal, Hinneburg and Keim then showed the rate depends on the norm. For uniform data under an $L_k$ metric, the expected gap between the farthest and nearest distances scales as $\lim_{d \to \infty} \mathbb{E}\!\left[ \frac{D_{\max}^{k} - D_{\min}^{k}}{d^{1/k - 1/2}} \right] = C \cdot \frac{1}{(k+1)^{1/k}} \sqrt{\frac{1}{2k+1}}$. with C constant, while typical distances themselves grow like $d^{1/k}$ . Relative contrast therefore decays with dimension, and the constant shrinks as k rises — which is why they conclude that $L_1$ is preferable to $L_2$ in high dimensions, $L_2$ to $L_3$ , and that fractional norms preserve contrast better still [ 12 ] .

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