Strongly self-dual polytopes

TL;DR

This paper explores the class of strongly self-dual polytopes, revealing new structures beyond Lovász's L-type polytopes, including a novel example with 23 vertices and classifications for polytopes with fewer than nine vertices.

Contribution

It identifies new strongly self-dual polytopes with different structures from known L-type polytopes and provides classification results for small vertex counts.

Findings

Existence of a 23-vertex ssd-polytope with different structure from L-type

In dimension three, faces of ssd-polytopes define L-type polyhedra

Only five ssd-polyhedra with fewer than nine vertices, four constructed by Lovász's method

Abstract

This article aims to study the class of strongly self-dual polytopes (ssd-polytopes for short), defined in a paper by Lov\'asz \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 Lov\'asz's method, and we can find the fifth one with "the diameter gradient flow algorithm" of Katz, Memoli and Wang \cite{katz-memoli-wang}.