2013/06/29 by Michael Gene Dobbins, Dobbins, Michael Gene
Computer Science · Engineering · Mathematics · #Advanced Combinatorial Mathematics #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG) #graph theory and CDMA systems #math.CO #math.MG
paper · pdf · doi:10.48550/arxiv.1307.0071
openalex publication_date 2013/06/29 · arxiv created 2015/06/22 · arxiv updated 2015/06/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This article exhibits a 4-dimensional combinatorial polytope that has no antiprism, answering a question posed by Bernt Lindstöm. 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.