Ordering Candidates via Vantage Points

Given an n-element set \(C\subseteq \mathbb {R}^d\) and a (sufficiently generic) k-element multiset \(V\subseteq \mathbb {R}^d\), we can order the points in C by ranking each point \(c\in C\) according to the sum of the distances from c to the points of V. Let \(\Psi _k(C)\) denote the set of orderings of C that can be obtained in this manner as V varies, and let \(\psi ^{\textrm{max}}_{d,k}(n)\) be the maximum of \(|\Psi _k(C)|\) as C ranges over all n-element subsets of \(\mathbb {R}^d\). We prove that \(\psi ^{\textrm{max}}_{d,k}(n)=\Theta _{d,k}(n^{2dk})\) when \(d \ge 2\) and that \(\psi ^{\textrm{max}}_{1,k}(n)=\Theta _k(n^{4\lceil k/2\rceil -2})\). 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 \(\Psi (C)=\bigcup _{k\ge 1}\Psi _k(C)\); this includes an exact description of \(\Psi (C)\) when \(d=1\) and when C is the set of vertices of a vertex-transitive polytope.