A Blichfeldt-type inequality for the surface area
| dc.creator | Henk, Martin | |
| dc.creator | Wills, Joerg M. | |
| dc.date | 2007-05-15 | |
| dc.date.accessioned | 2026-07-07T08:01:37Z | |
| dc.date.available | 2026-07-07T08:01:37Z | |
| dc.description | In 1921 Blichfeldt gave an upper bound on the number of integral points contained in a convex body in terms of the volume of the body. More precisely, he showed that $#(K\cap\Z^n)\leq n! \vol(K)+n$, whenever $K\subset\R^n$ is a convex body containing $n+1$ affinely independent integral points. Here we prove an analogous inequality with respect to the surface area $\F(K)$, namely $ #(K\cap\Z^n) < \vol(K) + ((\sqrt{n}+1)/2) (n-1)! \F(K)$. The proof is based on a slight improvement of Blichfeldt's bound in the case when $K$ is a non-lattice translate of a lattice polytope, i.e., $K=t+P$, where $t\in\R^n\setminus\Z^n$ and $P$ is an $n$-dimensional polytope with integral vertices. Then we have $#((t+P)\cap\Z^n)\leq n! \vol(P)$. Moreover, in the 3-dimensional case we prove a stronger inequality, namely $#(K\cap\Z^n) < \vol(K) + 2 \F(K)$. | |
| dc.identifier | https://arxiv.org/abs/0705.2088 | |
| dc.identifier | http://arxiv.org/abs/0705.2088 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/128965 | |
| dc.subject | Metric Geometry | |
| dc.subject | 52C07; 11H06 | |
| dc.title | A Blichfeldt-type inequality for the surface area | |
| dc.type | text |