The first eigenvector of a distance matrix is nearly constant

Abstract

Let x1,…,xn be points in a metric space and define the distance matrix D∈Rn×n by Dij=d(xi,xj). The Perron-Frobenius Theorem implies that there is an eigenvector v∈Rn with non-negative entries associated to the largest eigenvalue. We prove that this eigenvector is nearly constant in the sense that the inner product with the constant vector 1∈Rn is large〈v,1〉≥12⋅‖v‖ℓ2⋅‖1‖ℓ2 and that each entry satisfies vi≥‖v‖ℓ2/4n. Both inequalities are sharp.