Optimal randomized clustering for subsets of \(L_p\) when \(p \gt 2\)

We resolve multiple fundamental open questions about the bi-Lipschitz geometry of subsets of \(L_p\) for \(2 \le p \lt \infty\) via a novel multiscale and localization framework. Specifically, we prove that the separation modulus of any \(n\)-point subset of \(L_p\) is \(\Theta_p(\sqrt{\log n})\). If that subset has doubling constant \(\lambda\), then we obtain the improved bound \(O_p(\sqrt{\log \lambda})\), which is new even for the Euclidean space \(p = 2\). We also break the longstanding \(O(\log n)\) barrier for embedding every \(n\)-point subset of \(L_p\) into Euclidean space for all \(p \gt 2\), as well as the longstanding \(O((\log n)/\log\log n)\) barrier for their Lipschitz extension modulus.