Mutual Asymptotic Fekete Sequences

A sequence of point configurations on a compact complex manifold is asymptotically Fekete if it is close to maximizing a sequence of Vandermonde determinants. These Vandermonde determinants are defined by tensor powers of a Hermitian ample line bundle and the point configurations in the sequence possess good sampling properties with respect to sections of the line bundle. In this paper, given a collection of toric Hermitian ample line bundles, we give necessary and sufficient condition for existence of a sequence of point configurations which is asymptotically Fekete (and hence possess good sampling properties) with respect to each one of the line bundles. When they exist, we also present a way of constructing such sequences. As a byproduct we get a new equidistribution property for maximizers of products of Vandermonde determinants.