Press release · 8 September 2026 · version 1.0.0-candidate
A six-dimensional counterexample to cross-polytope maximality in zonoid approximation
A fourteen-vertex body in six dimensions has exact zonoid enclosure factor 122/65, exceeding 15/8 by 1/520.
Summary
The cross-polytope is a natural candidate for the symmetric shape hardest to enclose by sums of line segments and their limits. This release gives a counterexample: adding one antipodal pair to a six-dimensional cross-polytope increases the exact enclosure factor. The gain is small but strictly positive, and rational certificates prove it without relying on numerical tolerances.
Summary for specialists
The cross-polytope does not maximize the zonoid enclosure factor among all origin-symmetric convex bodies. This candidate gives a concrete counterexample in six dimensions:
\[ \begin{aligned} K&=\operatorname{conv}\{\pm v_1,\ldots,\pm v_7\},\\ v_j&=4e_j\quad(1\le j\le6),\\ v_7&=(1,1,1,1,2,2),\\ \lambda(K)&=\frac{122}{65}=\frac{15}{8}+\frac1{520}. \end{aligned} \]
Here \(\lambda(K)\) is the smallest factor \(t\) for which some zonoid lies between \(K\) and \(tK\). A zonoid is a Hausdorff limit of Minkowski sums of segments. The benchmark \(\lambda(C_6)=15/8\) is the value recorded in Schneider’s AIM question. The strict gap is small but exact; no floating-point tolerance enters the final certificate checks.
Technical account: why the finite witness covers every zonoid
The lower witness supplies 64 rational signed-sum identities. Negation covers the remaining sign choices. These identities imply a support inequality for every generator direction, not just a sampled list. Summing over generators proves it for zonotopes; continuity of support functions extends it to every centered zonoid. Symmetrization preserves the required enclosure, so allowing translated zonoids does not evade the bound.
The matching upper witness contains 53 rational generators and a \(7\times53\) coefficient matrix. Exact vertex representations prove the inner inclusion. Exact support checks at every facet prove the outer inclusion. A second enumeration describes the polar as a clipped cube: 54 retained cube vertices plus 16 new edge intersections give 70 polar vertices, or 35 antipodal facet pairs.
Evidence and replay
The linked paper contains the full argument and the nonzero lower weights. The evidence archive includes both rational certificates, standard-library Python checkers, historical discovery records and the review-response record.
python3 replay_review.py
python3 -O replay_review.py
python3 verify_polar.pyThe replay rejects eight corrupted lower witnesses and eight corrupted upper witnesses, including surplus and missing coordinates. The polar checker verifies all 70 upper-support equalities. These rejection controls test the checkers; they are not substitutes for the written proof or external validation.
What changed after review
The supplied review recommended minor revisions without identifying a fatal proof defect. This release adds explicit dimension checks, four corresponding rejection tests, a corrected accessible manuscript, related-work distinctions, the numerator calculation and the polar-body explanation. The supplied review’s separately linked audit ZIP was unavailable here, so its reported arithmetic checks are not counted as inspected independent reproduction.
Scope and limitations
This is an unrefereed computer-assisted candidate. Producer replay and five internal model-assisted editorial reports do not constitute unaffiliated specialist review, independent reproduction or formal verification. A bounded search for the final object found no matching earlier example, but does not clear historical priority.
The result does not identify the largest factor in dimension six, the smallest counterexample dimension, a unique enclosure or the minimum possible generator count. It provides no measured application benefit or research-speed comparison.
Relationship to earlier work
Schneider’s support-function arguments are antecedents of the method. Henk, Linke and Wills study enclosures of a fixed cross-polytope; Siegel studies finite approximation of bodies already known to be zonoids. Neither formulation should be conflated with maximizing the enclosure factor over arbitrary symmetric bodies.
Who should care, and why
| Audience | Potential use | Required caution |
|---|---|---|
| Convex geometers | An explicit test object for zonoid enclosure extremality | Not a classification of worst bodies |
| Optimization researchers | A rational dual witness with an all-direction interpretation | Discovery LP alone would not prove the lower bound |
| Proof-verification researchers | Small exact certificates and two facet checks | No proof-assistant formalization is supplied |
Why the problem matters
An extremal question asks more than whether one symmetric example is difficult. A single exact counterexample can separate a plausible benchmark from the actual extremal value. Here the matching enclosure also determines the new body's factor, not merely a lower estimate.
The most valuable next checks
The next useful assurance steps are unaffiliated scrutiny of the universal support-function argument, independent reconstruction of the rational certificates and a broader contribution-specific priority assessment. Historical research-goal metrics are separately labelled in the archive. The publication attempt below measures only its prospectively registered assurance-and-publication scope.
What is in the evidence package
The PDF and accessible Markdown give the full proof; the two root JSON certificates contain the rational witnesses. The replay scripts check identities, containments and malformed-input rejection. The separate polar checker explains the facet count. The archive also includes internal editorial reports, the supplied-review response, a bounded final-object search log, component licences and historical discovery records clearly separated from the authoritative root result.
Media
The audio briefing is provided in the header above. Download the MP3 briefing · read the transcript.
Open directions for follow-up research
Also available in machine-readable form for research agents and follow-up projects.
- What is the largest zonoid enclosure factor in dimension six?
- What is the smallest dimension admitting a counterexample?
- Are optimal enclosures unique, or can fewer than 53 generators suffice?
- Assess this particular example and proof against earlier literature independently.
Verification status
Unrefereed computer-assisted candidate. Exact producer replay and five internal model-assisted editorial reports are not unaffiliated specialist review, independent reproduction or formal verification. Historical priority, least counterexample dimension and global dimension-six extremality are not established.
Cite
BibTeX
@misc{sixdimensionalzonoidcounterexample2026,
title = {A six-dimensional counterexample to cross-polytope maximality in zonoid approximation},
author = {Anonymous},
year = {2026},
doi = {10.5281/zenodo.22660175},
url = {https://doi.org/10.5281/zenodo.22660175},
version = {1.0.0-candidate},
howpublished = {Zenodo},
note = {Unrefereed; internally replayed evidence package. Press page: https://evidencepress.org/releases/six-dimensional-zonoid-counterexample/}
}Also: cite.bib · paper.json · this page as Markdown