Manuscript En05-B@1 · En05-B.tex · lemma · 8b626460

Lemma: Hub metric bound

Uniform metric estimates on all sites · lines 586–608 · proof included or linked

En05-B@1#lem:hubmetric

For hubs $H_{k}, H_{k^{'}}$ ,

∥ z_{n} (H_{k}) - z_{n} (H_{k^{'}}) ∥_{2} \leq a_{n} d_{n} (H_{k}, H_{k^{'}}) \leq λ_{n} ∥ z_{n} (H_{k}) - z_{n} (H_{k^{'}}) ∥_{2} + s_{n} + a_{n} .

Proof. The lower bound follows by summing (eq:edgeLipschitz). For a route with $m \geq 1$ lanes, use one clique edge to enter the first lane, $N_{n}$ edges on each lane, one clique edge between consecutive lanes, and one clique edge to finish at the last hub. This uses

m N_{n} + (m - 1) + 2 = m F_{n} + 1

graph edges. Apply Lemma lem:hubroute. The equal-hub case is immediate.

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unlabeled · line 589
$∥ z_{n} (H_{k}) - z_{n} (H_{k^{'}}) ∥_{2} \leq a_{n} d_{n} (H_{k}, H_{k^{'}}) \leq λ_{n} ∥ z_{n} (H_{k}) - z_{n} (H_{k^{'}}) ∥_{2} + s_{n} + a_{n} .$
unlabeled · line 603
$m N_{n} + (m - 1) + 2 = m F_{n} + 1$

Lemma: Hub metric bound

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