2023/07/21 by Théophile Buffière, Buffière, Théophile, Lionel Pournin +1
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #Mathematics and Applications #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.2307.11366
openalex publication_date 2023/07/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A 3-dimensional polytope P is k-equiprojective when the projection of P along any line that is not parallel to a facet of P is a polygon with k vertices. In 1968, Geoffrey Shephard asked for a description of all equiprojective polytopes. It has been shown recently that the number of combinatorial types of k-equiprojective polytopes is at least linear as a function of k. Here, it is shown that there are at least k3k/2+o(k) such combinatorial types as k goes to infinity. This relies on the Goodman--Pollack lower bound on the number of order types and on new constructions of equiprojective polytopes via Minkowski sums.