Favourite distances in 3-space
Pith reviewed 2026-05-24 19:18 UTC · model grok-4.3
The pith
For large n any maximizer of favourite distances in 3-space lies on a circle and its axis except for at most two points.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
If the pair (S,r) maximises f_3(n) and n is sufficiently large, then, except for at most 2 points, S is contained in a circle C and the axis of symmetry L of C, and r(x) equals the distance from x to C for each x∈S∩L. This, together with a new construction, implies that f_3(n)=n²/4 + 5n/2 + O(1).
What carries the argument
The circle C together with its symmetry axis L, which together support all but at most two points of any large maximizer, with r(x) set to the distance from axis points to C.
If this is right
- f_3(n) equals n²/4 + 5n/2 plus a term bounded independently of n.
- Any maximizer has all but at most two points on one circle and its axis.
- The distance function on the axis points must be the common perpendicular distance to the circle.
- The quadratic term arises from the complete bipartite graph realized between the circle points and the axis points.
Where Pith is reading between the lines
- The same circle-axis support may be the only rigid structure that saturates the quadratic bound in three dimensions.
- Determining the precise constant inside the O(1) term would require a finer analysis of the small number of exceptional points.
- The result isolates the contribution of the axis points as a linear term of coefficient 5/2.
Load-bearing premise
Every maximizer for all sufficiently large n must satisfy the circle-plus-axis structural property except for at most two points.
What would settle it
A concrete point set of size n together with an assignment r whose total sum exceeds n²/4 + 5n/2 + C for some fixed C and arbitrarily large n, or a maximizer containing three or more points outside any single circle-axis pair.
read the original abstract
Let $S$ be a set of $n$ points in Euclidean $3$-space. Assign to each $x\in S$ a distance $r(x)>0$, and let $e_r(x,S)$ denote the number of points in $S$ at distance $r(x)$ from $x$. Avis, Erd\H{o}s and Pach (1988) introduced the extremal quantity $f_3(n)=\max\sum_{x\in S}e_r(x,S)$, where the maximum is taken over all $n$-point subsets $S$ of 3-space and all assignments $r\colon S\to(0,\infty)$ of distances. We show that if the pair $(S,r)$ maximises $f_3(n)$ and $n$ is sufficiently large, then, except for at most $2$ points, $S$ is contained in a circle $\mathcal{C}$ and the axis of symmetry $\mathcal{L}$ of $\mathcal{C}$, and $r(x)$ equals the distance from $x$ to $C$ for each $x\in S\cap\mathcal{L}$. This, together with a new construction, implies that $f_3(n)=n^2/4 + 5n/2 + O(1)$.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript studies the extremal quantity f_3(n), the maximum over n-point sets S in R^3 and assignments r:S→(0,∞) of the sum ∑_{x∈S} e_r(x,S), where e_r(x,S) counts points at distance r(x) from x. It proves that for all sufficiently large n, any maximizing pair (S,r) has the property that all but at most two points of S lie on a circle C together with its axis of symmetry L, and that r(x) equals the distance from x to C whenever x lies on L. Combined with an explicit construction, this yields the asymptotic f_3(n)=n²/4 + 5n/2 + O(1).
Significance. If the structural characterization holds, the paper determines the leading two terms in the asymptotic growth of f_3(n), thereby resolving the order of magnitude left open by Avis–Erdős–Pach (1988). The combination of a rigidity-type upper bound with a matching lower-bound construction is a standard and effective approach in extremal combinatorial geometry; the explicit form of the extremal configurations (circle plus axis) is a concrete contribution.
minor comments (2)
- The abstract and introduction should explicitly recall the definition of e_r(x,S) and f_3(n) before stating the structural theorem, to make the statement self-contained for readers who skip the preliminaries.
- Notation: the axis is denoted both by L and by script L in the abstract; a single consistent symbol should be used throughout.
Simulated Author's Rebuttal
We thank the referee for the positive assessment of the manuscript and the recommendation of minor revision. The report lists no major comments.
Circularity Check
No significant circularity
full rationale
The derivation consists of a structural characterization theorem for large-n maximizers of f_3(n) (S lies on a circle plus axis except for O(1) points, with r(x) the distance to the circle) proved via combinatorial geometry arguments on unit-distance graphs and extremal configurations, followed by an explicit new construction achieving the matching lower bound n²/4 + 5n/2 + O(1). The cited 1988 Avis-Erdős-Pach result is external prior work. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations appear; the upper-bound structure and lower-bound construction are independent of each other.
Axiom & Free-Parameter Ledger
axioms (1)
- standard math The ambient space is Euclidean 3-space equipped with the standard Euclidean distance.
Lean theorems connected to this paper
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IndisputableMonolith/Foundation/AlexanderDuality.leanalexander_duality_circle_linking echoes?
echoesECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.
Theorem B: if (S,r) maximises f_3(n) and n large then except ≤2 points, S contained in circle C and axis L, r(x)=dist(x,C) for x in S∩L
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IndisputableMonolith/Cost/FunctionalEquation.leanwashburn_uniqueness_aczel unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
er(S) ≤ ℓc + ℓ + 4c with ℓ=|L|, c=|C|; construction attaining ⌈n²/4 + 5n/2⌉ +1
What do these tags mean?
- matches
- The paper's claim is directly supported by a theorem in the formal canon.
- supports
- The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
- extends
- The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
- uses
- The paper appears to rely on the theorem as machinery.
- contradicts
- The paper's claim conflicts with a theorem or certificate in the canon.
- unclear
- Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.
discussion (0)
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