The first eigenvector of a distance matrix is nearly constant

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 largest eigenvalue. We prove this nearly constant sense inner product vector 1?Rn large?v,1??12??v??2??1??2 each entry satisfies vi??v??2/4n. Both inequalities are sharp.