A Blichfeldt-type inequality for the surface area

Details A Blichfeldt-type inequality for the surface area A Blichfeldt-type inequality for the surface area

2007-05-15

arxiv.org/abs/0705.2088v1

Mathematics

Metric Geometry

Scientific paper

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 $#(K\cap\Z^n) < \vol(K) + 2 \F(K)$.

Affiliated with

Also associated with

No associations

Rating

A Blichfeldt-type inequality for the surface area does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with A Blichfeldt-type inequality for the surface area, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and A Blichfeldt-type inequality for the surface area will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-622811