Ordering Candidates via Vantage Points

Abstract

Given an n-element set C ⊆ R d and a (sufficiently generic) k-element multiset V ⊆ R d , we can order the points in C by ranking each point c ∈ C according to the sum of the distances from c to the points of V. Let Ψ k ( C ) denote the set of orderings of C that can be obtained in this manner as V varies, and let ψ d , k max ( n ) be the maximum of | Ψ k ( C ) | as C ranges over all n-element subsets of R d . We prove that ψ d , k max ( n ) = Θ d , k ( n 2 d k ) when d ≥ 2 and that ψ 1 , k max ( n ) = Θ k ( n 4 ⌈ k / 2 ⌉ - 2 ) . As a step toward proving this result, we establish a bound on the number of sign patterns determined by a collection of functions that are sums of radicals of nonnegative polynomials; this can be understood as an analogue of a classical theorem of Warren. We also prove several results about the set Ψ ( C ) = ⋃ k ≥ 1 Ψ k ( C ) ; this includes an exact description of Ψ ( C ) when d = 1 and when C is the set of vertices of a vertex-transitive polytope.