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Strongly self-dual polytopes

Ákos G. Horváth, István Prok

TL;DR

This work broadens Lovász's theory of strongly self-dual polytopes by examining three-dimensional instances beyond the classic $L$-type family and by constructing a non-$L$-type ssd-polyhedron with $23$ vertices. It derives fundamental geometric and duality properties for ssd-polytopes, including a uniform diagonal length $\alpha=\sqrt{2+2r}$ and a duality map $\sigma$ that pairs vertices with orthogonal facets, and leverages these to constrain possible combinatorics. The paper further provides a constructive dual-edge framework via $\Phi_r$, enabling numerical exploration of new ssd-polyhedra from non-regular faces, and presents a concrete non-$L$-type example, together with an eight-vertex classification that confirms the finite nature of small SSD families. The results reveal a richer structural landscape for ssd-polytopes than previously recognized and offer algorithmic tools for discovering new instances with potential implications for self-polar and diameter-constrained configurations on the sphere.

Abstract

This article aims to study the class of strongly self-dual polytopes (ssd-polytopes for short), defined in a paper by Lovász \cite{lovasz}. He described a series of such polytopes (called $L$-type polytopes), which he used to solve a combinatorial problem. From a geometrical point of view, there are interesting questions: what additional elements of this class exist, and are there any with a different structure from the $L$-type ones? We show that in dimension three, one of their faces defines $L$-type polyhedra. Illustrating the algorithm of the proof, we present an ssd-polytope of 23 vertices whose combinatorial structure differ from those of $L$-type ones. Finally, with an elementary discussion, we prove that for fewer than nine vertices, there are only fifth one ssd-polyhedra, four of them can be constructed by Lovász's method, and we can find the fifth one with "the diameter gradient flow algorithm" of Katz, Memoli and Wang \cite{katz-memoli-wang}.

Strongly self-dual polytopes

TL;DR

This work broadens Lovász's theory of strongly self-dual polytopes by examining three-dimensional instances beyond the classic -type family and by constructing a non--type ssd-polyhedron with vertices. It derives fundamental geometric and duality properties for ssd-polytopes, including a uniform diagonal length and a duality map that pairs vertices with orthogonal facets, and leverages these to constrain possible combinatorics. The paper further provides a constructive dual-edge framework via , enabling numerical exploration of new ssd-polyhedra from non-regular faces, and presents a concrete non--type example, together with an eight-vertex classification that confirms the finite nature of small SSD families. The results reveal a richer structural landscape for ssd-polytopes than previously recognized and offer algorithmic tools for discovering new instances with potential implications for self-polar and diameter-constrained configurations on the sphere.

Abstract

This article aims to study the class of strongly self-dual polytopes (ssd-polytopes for short), defined in a paper by Lovász \cite{lovasz}. He described a series of such polytopes (called -type polytopes), which he used to solve a combinatorial problem. From a geometrical point of view, there are interesting questions: what additional elements of this class exist, and are there any with a different structure from the -type ones? We show that in dimension three, one of their faces defines -type polyhedra. Illustrating the algorithm of the proof, we present an ssd-polytope of 23 vertices whose combinatorial structure differ from those of -type ones. Finally, with an elementary discussion, we prove that for fewer than nine vertices, there are only fifth one ssd-polyhedra, four of them can be constructed by Lovász's method, and we can find the fifth one with "the diameter gradient flow algorithm" of Katz, Memoli and Wang \cite{katz-memoli-wang}.

Paper Structure

This paper contains 8 sections, 9 theorems, 47 equations, 19 figures.

Table of Contents

  1. Introduction
  2. Strongly self-dual three-dimensional polyhedra
  3. On the construction of Lovász
  4. Some further observations on ssd-polyhedra
  5. Ssd-polyhedra with up to eight vertices.
  6. The construction of a non-$L$-type ssd-polyhedron
  7. Formulas on the dual-edges
  8. Searching for a non-L-type ssd-polyhedron

Key Result

Theorem 1

The chromatic number of the graph formed by the principal diagonals of a strongly self-dual polytope in $\mathbb{R}^n$ is greater than or equal to $n+1$. Equality holds if its parameter is greater than the diameter of the monochromatic parts of the colouring defined by the regular simplex of dimensi

Figures (19)

  • Figure 1: The dual of a diagonal
  • Figure 2: Parallels and a meridian of $P(4,5)$.
  • Figure 3: The strongly self-dual polyhedra with at most eight vertices.
  • Figure 4: Two combinatorial cases of six vertices
  • Figure 5: Combinatorial cases of seven vertices
  • ...and 14 more figures

Theorems & Definitions (16)

  • Theorem 1: lovasz
  • Definition 1
  • Theorem 2: lovasz, ghorvath
  • Proposition 2.1
  • proof
  • Corollary 1
  • Proposition 2.2
  • proof
  • Corollary 2
  • Theorem 3
  • ...and 6 more