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arxiv: 2502.01878 · v1 · pith:KEWWCK7Ynew · submitted 2025-02-03 · 🧮 math.CO · math.OC

Determining inscribability of polytopes via rank minimization based on slack matrices

Yiwen Chen , Jo\~ao Gouveia , Warren Hare , Amy Wiebe This is my paper

Pith reviewed 2026-05-23 04:17 UTC · model grok-4.3

classification 🧮 math.CO math.OC
keywords polytope inscribabilityslack matrixrank minimizationsemidefinite programmingcombinatorial polytopessimplicial polytopesrealization problems
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The pith

A polytope is inscribable exactly when its slack matrix attains a minimum possible rank.

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

The paper gives a necessary and sufficient condition for a polytope to admit a realization with every vertex on a sphere. This condition is turned into an optimization task that asks for the lowest rank achievable by a slack matrix of the polytope. An SDP relaxation of the rank-minimization problem is introduced and shown to be exact on selected families of polytopes. Three algorithms are supplied whose size depends only on the number of vertices and facets, not on the ambient dimension. Experiments confirm that the method correctly classifies simplicial polytopes in dimensions four through eight with up to ten vertices.

Core claim

Inscribability holds if and only if the minimum rank of the slack matrix equals the value required by the polytope's combinatorial type; the authors therefore recast the decision problem as a rank-minimization task over slack matrices and supply an SDP surrogate that recovers the exact rank for the tested classes.

What carries the argument

Slack-matrix rank minimization, which encodes the geometric constraint that vertices lie on a sphere through the lowest attainable rank of a matrix derived from the face lattice.

If this is right

  • Inscribability checks become independent of the dimension of the polytope.
  • The SDP approximation is exact for simplicial polytopes with up to ten vertices in dimensions four to eight.
  • Three concrete algorithms are supplied that operate directly on the vertex-facet incidence data.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same slack-matrix formulation may be applied to other realization questions that can be expressed as low-rank conditions on incidence data.
  • If the SDP tightness extends beyond the tested range, enumeration of larger polytopes could become feasible without dimension-dependent blow-up.

Load-bearing premise

The SDP relaxation recovers the exact minimum rank for the polytopes on which the numerical tests are run.

What would settle it

A simplicial polytope in dimension five or higher whose SDP relaxation reports a rank strictly lower than the combinatorial minimum required by any inscribed realization.

Figures

Figures reproduced from arXiv: 2502.01878 by Amy Wiebe, Jo\~ao Gouveia, Warren Hare, Yiwen Chen.

Figure 1
Figure 1. Figure 1: Accuracy on random polytopes [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Accuracy on random polytopes [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
read the original abstract

A polytope is inscribable if there is a realization where all vertices lie on the sphere. In this paper, we provide a necessary and sufficient condition for a polytope to be inscribable. Based on this condition, we characterize the problem of determining inscribability as a minimum rank optimization problem using slack matrices. We propose an SDP approximation for the minimum rank optimization problem and prove that it is tight for certain classes of polytopes. Given a polytope, we provide three algorithms to determine its inscribability. All the optimization problems and algorithms we propose in this paper depend on the number of vertices and facets but are independent of the dimension of the polytope. Numerical results demonstrate our SDP approximation's efficiency, accuracy, and robustness for determining inscribability of simplicial polytopes of dimensions $4\le d\le 8$ with vertices $n\le 10$, revealing its potential in high dimensions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims a necessary-and-sufficient rank condition on the slack matrix of a polytope for inscribability (all vertices on a sphere). It reduces the decision problem to exact minimum-rank optimization over slack matrices, proposes an SDP relaxation, and proves the relaxation is tight for selected classes including simplicial polytopes. Three algorithms are given whose complexity depends only on the number of vertices and facets (independent of dimension). Numerical experiments test the SDP on simplicial polytopes for 4 ≤ d ≤ 8 and n ≤ 10 vertices.

Significance. If the rank characterization is correct and the SDP tightness proofs hold, the work supplies a new, dimension-independent computational test for inscribability that is potentially useful for high-dimensional polytopes where direct geometric methods scale poorly. The explicit SDP formulation and the reported tightness results for simplicial cases constitute concrete algorithmic contributions.

major comments (2)
  1. [§3, Theorem 3.1] §3, Theorem 3.1: the necessity direction of the rank condition is stated to follow from the existence of a spherical realization, but the argument that the resulting slack matrix must have rank exactly equal to the stated bound is only sketched; a complete derivation of the rank formula from the sphere equation is needed to confirm the equivalence is if-and-only-if rather than one-sided.
  2. [§4.3, Proposition 4.4] §4.3, Proposition 4.4: the proof that the SDP relaxation recovers the exact minimum rank for simplicial polytopes assumes a particular block structure of the slack matrix; it is not shown whether this structure is preserved under the combinatorial operations used to generate the test instances, which could affect the claimed tightness for the numerical regime 4 ≤ d ≤ 8.
minor comments (2)
  1. [§2] The notation for the slack matrix S(P) is introduced without an explicit reference to the standard definition in the literature; adding a short paragraph recalling the construction would improve readability.
  2. [Figure 2] Figure 2 caption does not state the dimension or number of vertices of the displayed polytope; this information should be added for reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful review and constructive comments. We address each major comment below, indicating revisions to be made.

read point-by-point responses

  1. Referee: [§3, Theorem 3.1] the necessity direction of the rank condition is stated to follow from the existence of a spherical realization, but the argument that the resulting slack matrix must have rank exactly equal to the stated bound is only sketched; a complete derivation of the rank formula from the sphere equation is needed to confirm the equivalence is if-and-only-if rather than one-sided.

    Authors: We agree that the necessity direction requires a fuller derivation. In the revised manuscript we will expand the proof of Theorem 3.1 to include an explicit, step-by-step derivation showing how the sphere equation implies that the slack matrix has rank exactly d+1 (combined with the known lower bound of d+1). This will confirm the if-and-only-if character of the rank characterization. revision: yes

  2. Referee: [§4.3, Proposition 4.4] the proof that the SDP relaxation recovers the exact minimum rank for simplicial polytopes assumes a particular block structure of the slack matrix; it is not shown whether this structure is preserved under the combinatorial operations used to generate the test instances, which could affect the claimed tightness for the numerical regime 4 ≤ d ≤ 8.

    Authors: The block structure used in the proof of Proposition 4.4 follows directly from the definition of the slack matrix for any simplicial polytope (zeros precisely on vertex-facet incidences). The combinatorial operations that generate the test instances preserve simpliciality and therefore preserve the incidence pattern and the resulting block form. We will insert a short clarifying remark after the statement of Proposition 4.4 to record this preservation, thereby confirming that the tightness result applies to the reported numerical regime. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper states a necessary-and-sufficient rank condition on slack matrices for inscribability (derived from the geometric definition of inscription), reduces the decision problem to exact minimum-rank optimization over those matrices, and separately proves SDP tightness for specified classes such as simplicial polytopes. These steps are presented as direct mathematical equivalences and independent proofs rather than reductions to fitted parameters, self-citations, or ansatzes. No load-bearing self-citation chains, self-definitional loops, or renaming of known results appear in the claims. The method is an optimization characterization derived from the slack-matrix representation, consistent with the reader's assessment of score 1.0.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract introduces no free parameters, additional axioms beyond standard convex geometry, or new postulated entities; the approach rests on the existing notions of slack matrices and rank minimization.

pith-pipeline@v0.9.0 · 5696 in / 1202 out tokens · 88219 ms · 2026-05-23T04:17:23.791904+00:00 · methodology

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Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages

  1. [1]

    H. H. Bauschke, D. R. Luke, H. M. Phan, and X. Wang , Restricted normal cones and the method of alternating projections: applications, Set-Valued and Variational Analysis, 21 (2013), pp. 475–501

    work page 2013

  2. [2]

    , Restricted normal cones and the method of alternating projections: theory, Set-Valued and Variational Analysis, 21 (2013), pp. 431–473

    work page 2013

  3. [3]

    H. H. Bauschke and W. M. Moursi , An introduction to convexity, optimization, and algorithms , Society for Industrial and Applied Mathematics, Philadelphia, PA, 2023

    work page 2023

  4. [4]

    Boyd and L

    S. Boyd and L. Vandenberghe, Convex optimization, Cambridge university press, 2004

    work page 2004

  5. [5]

    Braun, S

    G. Braun, S. Fiorini, S. Pokutta, and D. Steurer , Approximation limits of linear programs (beyond hierarchies), Mathematics of Operations Research, 40 (2015), pp. 756–772

    work page 2015

  6. [6]

    Eckart and G

    C. Eckart and G. Young , The approximation of one matrix by another of lower rank , Psychometrika, 1 (1936), pp. 211–218

    work page 1936

  7. [7]

    Fazel, Matrix rank minimization with applications , PhD thesis, Stanford University, 2002

    M. Fazel, Matrix rank minimization with applications , PhD thesis, Stanford University, 2002

    work page 2002

  8. [8]

    Fazel, H

    M. Fazel, H. Hindi, and S. Boyd , Rank minimization and applications in system theory , in Proceedings of the 2004 American control conference, vol. 4, IEEE, 2004, pp. 3273–3278

    work page 2004

  9. [9]

    Fiorini, S

    S. Fiorini, S. Massar, S. Pokutta, H. R. Tiwary, and R. De Wolf , Linear vs. semidefinite extended formulations: exponential separation and strong lower bounds , in Proceedings of the forty-fourth annual ACM symposium on Theory of computing, 2012, pp. 95–106

    work page 2012

  10. [10]

    Firsching, Realizability and inscribability for simplicial polytopes via nonlinear optimization , Mathemat- ical Programming, 166 (2017), pp

    M. Firsching, Realizability and inscribability for simplicial polytopes via nonlinear optimization , Mathemat- ical Programming, 166 (2017), pp. 273–295

    work page 2017

  11. [11]

    Gawrilow and M

    E. Gawrilow and M. Joswig , polymake: a framework for analyzing convex polytopes , Birkh¨ auser, Basel, 2000, pp. 43–73

    work page 2000

  12. [12]

    Gouveia, R

    J. Gouveia, R. Grappe, V. Kaibel, K. Pashkovich, R. Z. Robinson, and R. R. Thomas , Which nonneg- ative matrices are slack matrices? , Linear Algebra and its Applications, 439 (2013), pp. 2921–2933

    work page 2013

  13. [13]

    Gouveia, A

    J. Gouveia, A. Macchia, R. R. Thomas, and A. Wiebe , The slack realization space of a polytope , SIAM Journal on Discrete Mathematics, 33 (2019), pp. 1637–1653

    work page 2019

  14. [14]

    Gouveia, A

    J. Gouveia, A. Macchia, and A. Wiebe , Combining realization space models of polytopes , Discrete & Computational Geometry, 69 (2023), pp. 505–542

    work page 2023

  15. [15]

    , General non-realizability certificates for spheres with linear programming, Journal of Symbolic Com- putation, 114 (2023), pp. 172–192

    work page 2023

  16. [16]

    Gouveia, K

    J. Gouveia, K. Pashkovich, R. Z. Robinson, and R. R. Thomas, Four-dimensional polytopes of minimum positive semidefinite rank, Journal of Combinatorial Theory, Series A, 145 (2017), pp. 184–226. Determining inscribability of polytopes via rank minimization based on slack matrices 19

    work page 2017

  17. [17]

    Grant and S

    M. Grant and S. Boyd , Graph implementations for nonsmooth convex programs , in Recent Advances in Learning and Control, Lecture Notes in Control and Information Sciences, Springer-Verlag Limited, 2008, pp. 95–110. http://stanford.edu/~boyd/graph_dcp.html

    work page 2008

  18. [18]

    https://cvxr.com/cvx, 2014

    , CVX: Matlab software for disciplined convex programming, version 2.1. https://cvxr.com/cvx, 2014

    work page 2014

  19. [19]

    Gr¨unbaum, Convex polytopes, Graduate Texts in Mathematics, (2003)

    B. Gr¨unbaum, Convex polytopes, Graduate Texts in Mathematics, (2003)

    work page 2003

  20. [20]

    Lemon, A

    A. Lemon, A. M. So, and Y. Ye, Low-rank semidefinite programming: Theory and applications, Foundations and Trends® in Optimization, 2 (2016), pp. 1–156

    work page 2016

  21. [21]

    A. S. Lewis, D. R. Luke, and J. Malick , Local linear convergence for alternating and averaged nonconvex projections, Foundations of Computational Mathematics, 9 (2009), pp. 485–513

    work page 2009

  22. [22]

    Noll and A

    D. Noll and A. Rondepierre , On local convergence of the method of alternating projections , Foundations of Computational Mathematics, 16 (2016), pp. 425–455

    work page 2016

  23. [23]

    Padrol and G

    A. Padrol and G. M. Ziegler, Six topics on inscribable polytopes, Advances in discrete differential geometry, (2016), pp. 407–419

    work page 2016

  24. [24]

    Rivin , A characterization of ideal polyhedra in hyperbolic 3-space , Annals of mathematics, 143 (1996), pp

    I. Rivin , A characterization of ideal polyhedra in hyperbolic 3-space , Annals of mathematics, 143 (1996), pp. 51–70

    work page 1996

  25. [25]

    Rothvoß, The matching polytope has exponential extension complexity , Journal of the ACM, 64 (2017), pp

    T. Rothvoß, The matching polytope has exponential extension complexity , Journal of the ACM, 64 (2017), pp. 1–19

    work page 2017

  26. [26]

    Steiner, Systematische Entwicklung der Abh¨ angigkeit geometrischer Gestalten von einander, Fincke, 1832

    J. Steiner, Systematische Entwicklung der Abh¨ angigkeit geometrischer Gestalten von einander, Fincke, 1832

    work page

  27. [27]

    Steinitz, ¨Uber isoperimetrische probleme bei konvexen polyedern , Journal f¨ ur die reine und angewandte Mathematik, 159 (1928), pp

    E. Steinitz, ¨Uber isoperimetrische probleme bei konvexen polyedern , Journal f¨ ur die reine und angewandte Mathematik, 159 (1928), pp. 133–143

    work page 1928

  28. [28]

    Yannakakis, Expressing combinatorial optimization problems by linear programs , in Proceedings of the twentieth annual ACM symposium on Theory of computing, 1988, pp

    M. Yannakakis, Expressing combinatorial optimization problems by linear programs , in Proceedings of the twentieth annual ACM symposium on Theory of computing, 1988, pp. 223–228. A Eigenvalue computations A.1 Eigenvalues of M M⊤ in Subsubsection 3.2.1 Notice that M M⊤ = (n − 2)2 n2 cos4 π n   a b c · · · b b a b · · · c c b a · · · c ... ... ... .....

    work page 1988