The maximum number of points in the cross-polytope that form a packing set of a scaled cross-polytope

Ji Hoon Chun

REVIEW 2 major objections 5 minor 19 references

The maximum number of points in the cross-polytope that form a packing set of a scaled cross-polytope

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper pins down the largest sets of $\ell^1$-separated points in a unit cross-polytope: exactly $2n$ points at scales just below 1, and exactly 10 or 12 points in two three-dimensional ranges, with 14 as an upper bound below.

desk verdict Exact cross-polytope packing numbers are likely correct and worth a referee, but the printed lower-bound proofs assert the wrong inequality and the case table in Lemma 4.4 has errors. read the letter →

arxiv 1908.05650 v1 pith:5DVOD7NZ submitted 2019-08-15 math.MG

Ji Hoon Chun This is my paper · ORCID

classification math.MG MSC 52C17

keywords cross-polytopel1-distancepackingsetmaximumcardinalitythree-dimensionalkissingnumberexplicitconstruction

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper asks how many points can be placed inside the unit cross-polytope—the $\ell^1$ ball—so that every pair of distinct points is at least a given $\ell^1$-distance apart. It proves that in any dimension $n\ge 2$, when the required distance is close to the maximum possible, the answer is exactly the $2n$ vertices of the polytope. In three dimensions, it determines the exact maximum for two ranges of the distance parameter: 10 points when the scale lies in $(3/5,2/3]$ and 12 points when it lies in $(4/7,3/5]$, and it bounds the next range by 14 points. Each exact value is matched by an explicit set of points achieving it, so the bounds are sharp.

What carries the argument

The working mechanism is a division of the cross-polytope into $S_n(r)$, the part lying within $\ell^1$-distance $2r$ of a vertex, and, in three dimensions, eight leftover tetrahedra $\operatorname{conv}(V(r,\sigma))$ indexed by sign vectors. When $r>1-\frac{1}{n}$, the first part covers the whole body, and a uniqueness lemma shows each vertex can serve at most one packing point, forcing the $2n$ bound. For smaller $r$ in dimension 3, each leftover tetrahedron has $\ell^1$-diameter less than $2r$, so it holds at most one point; blocking lemmas then show that a point in one tetrahedron empties one or three neighbouring tetrahedra, which yields the upper bounds 10, 12, and 14 according to the interval.

What would settle it

Check the smallest pairwise $\ell^1$ distance in the ten-point set $V_3\cup Q_{10}$; if it falls below $4/3$, the construction fails for $r=2/3$ and Theorem 1.2(a) loses its lower bound. Similarly, an exhaustive search for an 11-point packing at $r=2/3$ would disprove the upper bound of 10.

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Extended reading notes

Core claim

The central discovery is a complete evaluation of the packing number $\gamma(C^*_n,r)$ on three intervals. Theorem 1.1 states that for $n\ge 2$ and $r\in(1-\frac{1}{n},1]$, $\gamma(C^*_n,r)=2n$, and that this interval is the largest on which the value remains $2n$. Theorem 1.2, for dimension 3, gives $\gamma(C^*_3,r)=10$ when $r\in(3/5,2/3]$, $\gamma(C^*_3,r)=12$ when $r\in(4/7,3/5]$, and $\gamma(C^*_3,r)\le 14$ when $r\in(1/2,4/7]$. Proposition 1.3 supplies a 13-point packing for $r\in(1/2,6/11]$, improving the lower bound within the last interval. The upper bounds come from a geometric decomposition into regions of diameter below $2r$, and the lower bounds come from explicit coordinate lists.

Load-bearing premise

The paper's lower-bound constructions must genuinely be packing sets: every pair of listed points has to be separated by at least the claimed $\ell^1$ distance, and the proofs state the inequality in the wrong direction (an upper bound where a lower bound is needed), although the coordinates themselves satisfy the intended condition.

Editorial extensions

If this is right

For every $n\ge 2$, any packing with $r\in(1-\frac{1}{n},1]$ has at most $2n$ points, and the vertex set achieves this bound, so the vertices are optimal throughout this interval.
In three dimensions, the maximum cardinality is exactly 10 on $(3/5,2/3]$ and exactly 12 on $(4/7,3/5]$, so no intermediate maximums occur in these ranges.
For $r\in(1/2,4/7]$ the maximum is at most 14, and at least 12 everywhere; the 13-point construction for $r\in(1/2,6/11]$ shows the lower bound is 13 on that subinterval.
Taking $r\to 0$, the scaled packing numbers $\gamma(C^*_n,r)$ are linked to the packing density of the cross-polytope, and the finite-$r$ results provide the first terms of that relationship.
The interval $(1-\frac{1}{n},1]$ is maximal: for $r\le 1-\frac{1}{n}$ one can place $2n+2$ points by adding the centroids of two opposing facets to the vertices.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same vertex-neighbourhood decomposition may extend to dimensions $n\ge 4$: when the leftover regions have $\ell^1$-diameter below $2r$, a similar blocking argument could yield exact values for other intervals of $r$.
The manuscript's lower-bound proofs state pairwise distances as upper bounds ($\le 4/3$, $\le 6/5$, $\le 12/11$) where the packing condition requires lower bounds; direct computation of the listed coordinates shows the minimal distances equal those values, so the constructions are valid but the text appears to contain a sign typo.
The gap between 13 and 14 for $r\in(6/11,4/7]$ suggests a possible 14-point configuration may exist near $r=6/11$; the paper's local argument rules out one natural family, but a different arrangement might close the gap.
At $r=1/2$, the bounds $19\le\gamma(C^*_3,1/2)\le26$ could be tightened by combining the decomposition ideas here with the known kissing structure, potentially resolving a piece of the cross-polytope kissing problem.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: the exact values are probably right, but the printed lower-bound proofs do not prove what they claim. The upper-bound part is the real meat and looks largely sound. This is a fixable paper, not a broken one.

What is new: Theorem 1.1 gives the sharp n-dimensional cutoff gamma(C*_n,r)=2n for r in (1-1/n,1]; Theorem 1.2 gives exact 3D values 10 and 12 on two intervals and an upper bound 14 on a third; Proposition 1.3 gives 13 points. The blocking-set method, adapted from Larman-Zong and Boroczky-Wintsche, is appropriate, and the upper-bound lemmas (2.1, 3.1-3.3, 4.1, 4.2, 4.6) are detailed and convincing. The connection to kissing numbers and packing density is honestly placed; the references look right and there is no self-citation inflation.

Soft spots: Propositions 5.1-5.4 all state ||x-y||_1 <= 4/3 (or 6/5, 12/11) where the packing condition requires >=. As written, none of those proofs verifies the packing property. I checked the coordinate lists; the true minimum distances are exactly those values, so the constructions are valid and the defect is a sign typo repeated four times. Still, it must be fixed. Lemma 4.4 has a second defect: several entries in the 20-case table are wrong as printed (case 10 gives r, but the vectors give 2-2r; case 17 gives 4/3-2r instead of 2-2r). Recomputing all 20 cases shows every distance is below 2r for the stated r-range, so the lemma's conclusion is true, but the table cannot stand. There are also small cross-reference errors (e.g., 'Theorem 1.3' vs 'Proposition 1.3'; Lemma 4.3 used in place of Lemma 4.2/4.4). All are minor and repairable.

Who it is for: people working on translative packing numbers, kissing numbers of polytopes, or finite packings in l1 geometry. It is a narrow but genuine exact computation. The paper also honestly flags its limits, including the unresolved interval (1/2,4/7] and the possibility that the 13-point configuration might be improved.

Recommendation: send to peer review. The mathematical claims appear correct and the method is sound; the written proof needs a careful check and a corrected lower-bound section before acceptance.

Referee Report

2 major / 5 minor

Summary. The paper studies gamma(C*_n,r), the maximum size of a subset D of the unit cross-polytope C*_n whose distinct points are pairwise at l1-distance at least 2r. It proves Theorem 1.1, that gamma(C*_n,r)=2n for r in (1-1/n,1], and Theorem 1.2, that in dimension three gamma(C*_3,r)=10 for r in (3/5,2/3], gamma(C*_3,r)=12 for r in (4/7,3/5], and gamma(C*_3,r)<=14 for r in (1/2,4/7]. The upper bounds are obtained by decomposing C*_3 into the vertex neighborhoods S_3(r) and eight regions conv(V(r,(sigma_1,sigma_2,sigma_3))), then showing that certain of these regions are blocked sets. The lower bounds are explicit configurations: V3 union Q10, V3 union Q+12 union -Q+12, and V3 union Q13, giving respectively 10, 12, and 13 points for the stated ranges of r. The paper also relates the problem to the packing density of the cross-polytope and to kissing numbers.

Significance. If the results are correct, they provide exact maximum cardinalities for finite l1 packings in cross-polytopes, a problem that complements known results on kissing numbers and on the quantities M(L,K,m). The upper-bound architecture is largely self-contained and convincing: Lemma 2.1 gives a useful containment, Lemmas 3.2 and 3.3 control the number of points near vertices, and the blocked-set arguments in Section 4 are conceptually sound. The explicit coordinate constructions are a strength and appear to be genuine packing sets. However, the printed proofs of the lower bounds contain reversed inequalities, and the proof of Lemma 4.4 contains arithmetically incorrect case computations, so the exact statements are not established by the manuscript as written. Direct recomputation of the listed configurations indicates that the intended inequalities hold, so a substantive revision should be able to repair the paper rather than change its conclusions.

major comments (2)

[Section 5, Propositions 5.1–5.4] The lower-bound proofs assert upper bounds on pairwise distances, which is the opposite of what the packing condition requires. For example, Proposition 5.2 states that any distinct x,y in V3 union Q10 satisfy ||x-y||_1 <= 4/3, but to be a packing set for r<=2/3 one needs ||x-y||_1 >= 2r, and in particular a lower bound of at least 4/3 at the endpoint r=2/3. The same reversed inequality appears in Proposition 5.3 as ||x-y||_1 <= 6/5 and in Proposition 5.4 as ||x-y||_1 <= 12/11, and Proposition 5.1 has the analogous statement ||x-y||_1 <= 2(1-1/n). As printed, these propositions do not verify the packing property and therefore do not establish the lower-bound halves of Theorem 1.2 or Proposition 1.3. Direct enumeration of the listed sets shows that their true minimum distances are exactly 4/3, 6/5, and 12/11 respectively, so replacing the upper bounds by lower bounds and checking the endpoint equalities repairs the proofs.
[Section 4.3, Lemma 4.4] The 20-case table contains several false computations, and since Lemma 4.4 is load-bearing for the upper bound gamma(C*_3,r)<=12 in Theorem 1.2(b), these errors must be corrected. For instance, in case 10 the two displayed vectors differ by (r/2+1-r, 0, r/2-(2r-1)), whose l1-distance simplifies to 2-2r, not to r as printed; in case 17 the distance between (1/3,1/3,1/3) and (-(2r-1),2r-1,2r-1) is 2-2r, not 4/3-2r; and case 18 similarly miscomputes the first coordinate. These incorrect intermediate equalities do not immediately invalidate the lemma because the correct values are still below 2r on r in (4/7,3/5], but the written proof is not sound as it stands. The author should recompute all 20 cases and replace the erroneous expressions.

minor comments (5)

[Throughout Section 4] Several lemma references are mismatched: the proof of Lemma 4.3 cites 'Lemma 4.3' where Lemma 4.2 is meant; the proof of Theorem 1.2(a) cites Lemma 4.4 where Lemma 4.3 is meant; the proof of Theorem 1.2(b) cites Lemma 4.6 where Lemma 4.5 is meant; and the proof of Theorem 1.2(c) cites Lemma 4.5 where Lemma 4.6 is meant. These should be corrected to avoid confusing the reader.
[Proposition 5.1] The statement says Vn union {+-qn} subset C*_3, but this is the n-dimensional cross-polytope and should read C*_n; the same typo appears in the proof of Propositions 5.2 and 5.3 where C*_n is used instead of C*_3.
[End of Section 2] The text says 'Theorem 1.3 is proved in Section 5', but the referenced result is Proposition 1.3, not Theorem 1.3.
[Section 4.4 and proofs of Theorems 1.2(b) and 1.2(c)] The notation S*_3 appears in the splitting argument of the proofs of Theorem 1.2(b) and (c), but only S_3 and C*_3 are defined in Section 4.1. Use C*_3\S_3(r) consistently.
[Lemma 4.2 proof] The proof writes 'p in V(r,(1,1,1))' but the hypothesis is p in conv(V(r,(1,1,1))). The intended convexity argument should be stated explicitly, using the fact that the l1 norm is convex.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the packings and upper bounds are derived self-contained from definitions and explicit coordinate sets; only proof-typo correctness issues appear.

full rationale

The paper's derivation is not circular. Theorem 1.1 follows from Lemma 3.1 (C*_n = S_n(r) for r > 1-1/n), Lemma 3.2 (uniqueness near a vertex), and Lemma 3.3 (|P_n(r) ∩ S_n(r)| ≤ 2n), with the lower bound given by the vertex set V_n — all self-contained geometric arguments. The three-dimensional upper bounds in Theorem 1.2 are obtained from Lemmas 4.1–4.6, which analyze the regions conv(V(r,(σ1,σ2,σ3))) directly from the definition of l1-distance; no fitted parameter, normalization, or prior result of the author is invoked. The lower bounds are explicit coordinate sets (Propositions 5.1–5.4 and Q13), and although those propositions state inequalities such as '||x−y||_1 ≤ 4/3' where the packing definition requires ||x−y||_1 ≥ 2r — an inequality-direction typo — the listed sets are concrete and can be checked directly, so the lower-bound claims do not reduce to the theorem statements. External citations (Hadwiger, Larman–Zong, Talata, Swinnerton-Dyer) supply background kissing-number bounds and are not load-bearing for the main derivation. There is no self-citation chain, no renamed known result, and no uniqueness theorem imported from the author's own work. The appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper is a self-contained geometric derivation. It introduces no free parameters, no new geometric entities, and no ad hoc assumptions beyond standard convex-geometry facts. All lemmas are proved from the definitions, and the lower-bound constructions are explicit coordinate lists.

assumptions (3)

standard math Definition of translative packing and the gauge norm as used in gamma(L,K,r).
The problem is defined through this standard packing formalism; no alternative interpretation is needed.
standard math Facts about convex hulls, vertices, and sign-orthant decompositions of the cross-polytope.
The geometric decompositions in Section 4 rely on elementary convex geometry that is proved or standard.
standard math Finite counting and pigeonhole arguments in the blocked-set lemmas.
The upper-bound proofs use simple cardinality comparisons between vertex sets and packing points; no nonstandard combinatorial assumption is introduced.

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Pith. "Pith review of The maximum number of points in the cross-polytope that form a packing set of a scaled cross-polytope." pith.science (2026). https://pith.science/paper/5DVOD7NZ

@misc{pith2026190805650,
  author       = {Pith},
  title        = {Pith review of: The maximum number of points in the cross-polytope that form a packing set of a scaled cross-polytope},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5DVOD7NZ}},
  note         = {Machine review of arXiv:1908.05650}
}

abstract

The problem of finding the largest number of points in the unit cross-polytope such that the $l_{1}$-distance between any two distinct points is at least $2r$ is investigated for $r\in\left(1-\frac{1}{n},1\right]$ in dimensions $\geq2$ and for $r\in\left(\frac{1}{2},1\right]$ in dimension $3$. For the $n$-dimensional cross-polytope, $2n$ points can be placed when $r\in\left(1-\frac{1}{n},1\right]$. For the three-dimensional cross-polytope, $10$ and $12$ points can be placed if and only if $r\in\left(\frac{3}{5},\frac{2}{3}\right]$ and $r\in\left(\frac{4}{7},\frac{3}{5}\right]$ respectively, and no more than $14$ points can be placed when $r\in\left(\frac{1}{2},\frac{4}{7}\right]$. Also, constructive arrangements of points that attain the upper bounds of $2n$, $10$, and $12$ are provided, as well as $13$ points for dimension $3$ when $r\in\left(\frac{1}{2},\frac{6}{11}\right]$.

Figures

Figures reproduced from arXiv: 1908.05650 by the authors.

**Figure 4.** [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

**Figure 6.** [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗

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Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Böröczky Jr, I

K. Böröczky Jr, I. Fábián, and G. Wintsche , Covering the crosspolytope by equal balls, Periodica Mathematica Hungarica - PERIOD MATH HUNG, 53 (2006), pp. 103–113

work page 2006

[2] [2]

J. H. Conw ay and N. J. A. Sloane , Sphere Packings, Lattices and Groups, vol. 290 of Grundlehren der mathematischen Wissenschaften, Springer-Verlag New York, Inc., 3rd ed., 1998

work page 1998

[3] [3]

H. S. M. Coxeter , Regular Polytopes, Methuen & Co. Ltd., London, 1948

work page 1948

[4] [4]

H. T. Croft, K. J. F alconer, and R. K. Guy , Unsolved Problems in Geometry, vol. 2 of Problem books in mathematics, Unsolved problems in intuitive mathematics, Springer Science+Business Media New York, 1991

work page 1991

[5] [5]

L. Fu, W. Steinhardt, H. Zhao, J. E. S. Socolar, and P. Charbonneau , Hard sphere packings within cylinders, Soft Matter, 12 (2016), pp. 2505–2514

work page 2016

[6] [6]

J. E. Goodman, J. O’Rourke, and C. Tóth , Handbook of Discrete and Computational Geometry, Discrete Mathematics and Its Applications, Taylor & Francis Group, 3rd ed., 2018

work page 2018

[7] [7]

P. M. Gruber , Convex and Discrete Geometry, vol. 336 of Grundlehren der mathematischen Wis- senschaften, Springer-Verlag Berlin Heidelberg, 2007

work page 2007

[8] [8]

P. M. Gruber and C. G. Lekkerkerker , Geometry of Numbers, vol. 37 of North-Holland mathe- matical library, Elsevier Science Publishers B.V., 2nd ed., 1987

work page 1987