Antiprismless, or: Reducing Combinatorial Equivalence to Projective Equivalence in Realizability Problems for Polytopes

TL;DR

This paper constructs a 4-dimensional polytope without an antiprism, demonstrating a method to reduce combinatorial realizability problems to projective equivalence, with implications for understanding polytope realizations.

Contribution

It introduces a new approach to solving realizability problems by linking combinatorial properties to projective equivalence, and constructs a novel stamp polytope related to universality proofs.

Findings

Existence of a 4D polytope with no antiprism

Reduction of realizability problems to projective equivalence

Construction of a new stamp polytope related to universality

Abstract

This article exhibits a 4-dimensional combinatorial polytope that has no antiprism, answering a question posed by Bernt Lindst\"om. As a consequence, any realization of this combinatorial polytope has a face that it cannot rest upon without toppling over. To this end, we provide a general method for solving a broad class of realizability problems. Specifically, we show that for any semialgebraic property that faces inherit, the given property holds for some realization of every combinatorial polytope if and only if the property holds from some projective copy of every polytope. The proof uses the following result by Below. Given any polytope with vertices having algebraic coordinates, there is a combinatorial "stamp" polytope with a specified face that is projectively equivalent to the given polytope in all realizations. Here we construct a new stamp polytope that is closely related to Richter-Gebert's proof of universality for 4-dimensional polytopes, and we generalize several tools from that proof.