The center of distances of a finite ultrametric space: centered spheres, prescribed centers, and extremal rigidity

Abstract

The center of distances of a metric space (X,d) is the set C(X) of those distances t for which the equation d(p,x)=t has a solution x for every p ∈ X. Dovgoshey and Rovenska (arXiv:2601.13363, Mathematics 2026) described C(X) for ultrametric spaces generated by labeled trees and closed with three conjectures; their follow-up (arXiv:2603.25850) proved the third one, |C(X)| ≤ 1+flog(n) for every n-point ultrametric space with equality attainable for every n, and posed three more. We settle four of the five remaining conjectures and half of the fifth. Conjecture 2 holds: every non-empty subset of an ultrametric space with at least three points is a centered sphere if and only if the space has exactly three points and is weakly similar to the non-equidistant three-point space. The first half of Conjecture 1 holds: if every closed ball is a centered sphere then every truncated distance set Dₚ(X) ∩ [0,r] has a greatest element. Conjecture 4.4 holds: every finite set Ani 0 of non-negative reals is simultaneously the distance set and the center of distances of some finite ultrametric space, which can be taken to have 2^(|A|-1) points, the least possible. For the extremal spaces, |X|=2ⁿ with |C(X)|=n+1, we prove that every distance is centered, D(X)=C(X), and that every closed ball has exactly 2ᵏ points where k is the number of positive centered distances not exceeding its radius; a partition-count argument, presented on paper, then identifies every extremal space with the complete binary hierarchy and yields Conjectures 4.2 and 4.3. We also give an independent proof of the logarithmic bound through a single ball-doubling lemma, with a new explicit family attaining it, and record the controls: the bound fails for the 7-cycle metric, the +1 cannot be dropped, and an exhaustive check over all three-point tables agrees. Everything except the identification of the extremal spaces and an exhaustive enumeration to n ≤ 9 is verified in Lean 4 against Mathlib, without compiled evaluation.

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