On the Hadwiger covering problem in low dimensions
- PDF / 353,562 Bytes
- 11 Pages / 439.37 x 666.142 pts Page_size
- 70 Downloads / 215 Views
Journal of Geometry
On the Hadwiger covering problem in low dimensions A. Prymak
and V. Shepelska
Abstract. Let Hn be the minimal number of smaller homothetic copies of an n-dimensional convex body required to cover the whole body. Equivalently, Hn can be defined via illumination of the boundary of a convex body by external light sources. The best known upper bound in threedimensional case is H3 ≤ 16 and is due to Papadoperakis. We use Papadoperakis’ approach to show that H4 ≤ 96, H5 ≤ 1091 and H6 ≤ 15373 which significantly improve the previously known upper bounds on Hn in these dimensions. Mathematics Subject Classification. Primary 52A20; Secondary 52A37, 52A40, 52C17. Keywords. Illumination problem, Illumination number, Covering number, Covering by smaller homothetic copies, Convex body, 4-cube.
1. Introduction and results Let En denote the n-dimensional Euclidean space. A convex body in En is a convex compact set having non-empty interior. For two sets A, B ⊂ En we let C(A, B) be the smallest number of translates of B required to cover A, and let int(A) denote the interior of A. Hadwiger [3] asked what is the smallest value Hn of C(K, int(K)) for arbitrary convex body K in En . This is equivalent to the question about the least number of smaller homothetic copies of K which are able to cover K, and, as was shown by Boltyanski [2], to the question about the smallest number of external light sources required to illuminate the boundary of every convex body. Considering cube, one immediately gets Hn ≥ 2n . The related primary conjecture, which is commonly referred to as Hadwiger conjecture or as Gohberg–Markus covering A. Prymak was supported by NSERC of Canada Discovery Grant RGPIN-2020-05357. V. Shepelska was partially supported by the PIMS Postdoctoral Fellowship. 0123456789().: V,-vol
42
Page 2 of 11
A. Prymak, V. Shepelska
J. Geom.
conjecture, is that Hn = 2n , but this is known (and is simple) only for n = 2. Below we give a brief overview of the known results about Hn . For a detailed history of the question and survey including many partial results for special classes of convex bodies see, e.g., [1]. The best known explicit upper bound on Hn in high dimensions is a combination of Rogers’s result [6] on the covering density of En by translates of arbitrary convex body with the Rogers–Shephard inequality [7] bounding the volume of the difference body. The interested reader is referred to [1, Section 2.2] for further details and related results. Here we only state the actual bound, which is 2n n(ln n + ln ln n + 5), (1.1) Hn ≤ n where 5 can be replaced by 4 for sufficiently large n. Lassak [4] showed that Hn ≤ (n + 1)nn−1 − (n − 1)(n − 2)n−1 ,
(1.2)
which is better than (1.1) for n ≤ 5 and up to now was the best known bound for n = 4, 5. In [5] Papadoperakis showed that H3 ≤ 16, which is the best known bound in three dimensions. The key idea of [5] is to reduce the problem of Hadwiger to that of covering specific sets of relatively simple structure by certain rectangular parallelotopes. Namely, w
Data Loading...
