Ordering Candidates via Vantage Points
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Author(s)
Alon, Noga
Defant, Colin
Kravitz, Noah
Zhu, Daniel G.
Date Issued
April 8, 2025
Journal
Combinatorica
Publisher
Springer Berlin Heidelberg
Citation
Alon, N., Defant, C., Kravitz, N. et al. Ordering Candidates via Vantage Points. Combinatorica 45, 25 (2025).
Version
Author's final manuscript
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s00493-025-00148-0
