A polylogarithmic higher-order Cheeger inequality

Let $λ_k(G)$ be the $k$th eigenvalue of the normalized Laplacian of a finite undirected weighted graph, and let $ρ_G(k)$ be the minimum possible maximum conductance of $k$ disjoint nonempty vertex sets. We prove \[ ρ_G(k)\le C[1+\log(k+1)]^5\sqrt{λ_k(G)} \] for an absolute constant $C$. The constr