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Determining inscribability of polytopes via rank minimization based on slack matrices

Yiwen Chen, João Gouveia, Warren Hare, Amy Wiebe

TL;DR

This work develops a slack-matrix framework to decide polytope inscribability, proving a necessary and sufficient condition that translates into a minimum rank optimization problem. A convex SDP relaxation is proposed and shown to be tight for several polytope families, enabling efficient certification of inscribability via rank-$d{+}1$ witnesses. To solve the optimization in practice, the authors introduce three algorithms—an SDP solver and two rank-reduction methods based on alternating projection—with a simplified variant to speed up convergence. Numerical experiments on simplicial polytopes in dimensions $4\le d\le 8$ with up to $n\le 10$ vertices demonstrate that the SDP approach is robust and fast, while the AP-based methods offer higher accuracy at a cost in runtime. The framework is dimension-agnostic, offering a practical tool for high-dimensional inscribability tests and suggesting future directions for non-inscribability certificates and improved parameter tuning.

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.

Determining inscribability of polytopes via rank minimization based on slack matrices

TL;DR

This work develops a slack-matrix framework to decide polytope inscribability, proving a necessary and sufficient condition that translates into a minimum rank optimization problem. A convex SDP relaxation is proposed and shown to be tight for several polytope families, enabling efficient certification of inscribability via rank- witnesses. To solve the optimization in practice, the authors introduce three algorithms—an SDP solver and two rank-reduction methods based on alternating projection—with a simplified variant to speed up convergence. Numerical experiments on simplicial polytopes in dimensions with up to vertices demonstrate that the SDP approach is robust and fast, while the AP-based methods offer higher accuracy at a cost in runtime. The framework is dimension-agnostic, offering a practical tool for high-dimensional inscribability tests and suggesting future directions for non-inscribability certificates and improved parameter tuning.

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 with vertices , revealing its potential in high dimensions.

Paper Structure

This paper contains 28 sections, 12 theorems, 59 equations, 2 figures, 6 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Notation
  3. Characterizing inscribability using slack matrices
  4. SDP approximation
  5. Problem formulation
  6. Examples where the SDP approximation is accurate
  7. Example 1: $n$-gons
  8. Example 2: Simplices
  9. Example 3: Cubes
  10. Example 4: Cross-polytopes
  11. Algorithms
  12. SDP
  13. Alternating projection
  14. Simplified alternating projection
  15. Numerical experiments
  16. ...and 13 more sections

Key Result

lemma thmcounterlemma

Projective transformations preserve the combinatorial type of a polytope and the support of its slack matrix.

Figures (2)

  • Figure 1: Accuracy on random polytopes
  • Figure 2: Accuracy on random polytopes

Theorems & Definitions (19)

  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof
  • theorem 1
  • proof
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • proof
  • theorem 2
  • ...and 9 more