A sharp higher-order Cheeger inequality

Let $λ_k(G)$ be the $k$th eigenvalue of the normalized Laplacian of a finite undirected weighted graph $G$ with positive degrees, where $k$ is an integer satisfying $1\le k\le |V(G)|$. Let $φ_k(G)$ be the minimum possible maximum conductance of $k$ disjoint nonempty vertex sets. We prove $φ_k(G)\le