Plurality in Spatial Voting Games with constant $β$.

Consider a set V of voters, represented by a multiset in a metric space (X,d). The voters have to reach a decision -- a point in X. A choice p∈X is called a β-plurality point for V, if for any other choice q∈X it holds that |{v∈V∣β⋅d(p,v)≤d(q,v)}|≥|V|2. In other words, at least half of the voters ``prefer'' p over q, when an extra factor of β is taken in favor of p. For β=1, this is equivalent to Condorcet winner, which rarely exists. The concept of β-plurality was suggested by Aronov, de Berg, Gudmundsson, and Horton [TALG 2021] as a relaxation of the Condorcet criterion.
Let β∗(X,d)=sup{β∣every finite multiset V in X admits a β-plurality point}. The parameter β∗ determines the amount of relaxation required in order to reach a stable decision. Aronov et al. showed that for the Euclidean plane β∗(R2,∥⋅∥2)=3√2, and more generally, for d-dimensional Euclidean space, 1d√≤β∗(Rd,∥⋅∥2)≤3√2. In this paper, we show that 0.557≤β∗(Rd,∥⋅∥2) for any dimension d (notice that 1d√<0.557 for any d≥4). In addition, we prove that for every metric space (X,d) it holds that 2–√−1≤β∗(X,d), and show that there exists a metric space for which β∗(X,d)≤12.