2024/03/31 by Andrii Arman, Andriy Bondarenko, Arman, Andrii +3 · 1 citation
Computer Science · Mathematics · #52A20 #52A39 #52A40 #52C07 #52C17 #Computational Geometry and Mesh Generation #FOS: Mathematics #Geometric and Algebraic Topology #Mathematics and Applications #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.2404.00547
openalex publication_date 2024/03/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Hn be the minimal number such that any n-dimensional convex body can be covered by Hn translates of interior of that body. Similarly Hns is the corresponding quantity for symmetric bodies. It is possible to define Hn and Hns in terms of illumination of the boundary of the body using external light sources, and the famous Hadwiger's covering conjecture (illumination conjecture) states that Hn=Hns=2n. In this note we obtain new upper bounds on Hn and Hns for small dimensions n. Our main idea is to cover the body by translates of John's ellipsoid (the inscribed ellipsoid of the largest volume). Using specific lattice coverings, estimates of quermassintegrals for convex bodies in John's position, and calculations of mean widths of regular simplexes, we prove the following new upper bounds on Hn and Hns: H5≤ 933, H6≤ 6137, H7≤ 41377, H8≤ 284096, H4s≤ 72, H5s≤ 305, and H6s≤ 1292. For larger n, we describe how the general asymptotic bounds Hn≤ \binom2nnn(ln n+lnln n+5) and Hns≤ 2n n(ln n+lnln n+5) due to Rogers and Shephard can be improved for specific values of n.