by Bey, C., Henk, M., Wills, J. M..

**Series:** 2006-30, Preprints

- MSC:
- 52C07 Lattices and convex bodies in $n$ dimensions
- 11H06 Lattices and convex bodies

**Abstract:**

We determine lattice polytopes of smallest volume with a given

number of interior lattice points. We show that the Ehrhart

polynomials of those with one interior lattice point have

largest roots with norm of order $n^2$, where $n$ is the

dimension. This improves on the previously best known bound

$n$ and complements a recent result of Braun [8] where it is

shown that the norm of a root of a Ehrhart polynomial is at

most of order $n^2$.

For the class of 0-symmetric lattice polytopes we present

a conjecture on the smallest volume for a given number of

interior lattice points and prove the conjecture for

crosspolytopes.

We further give a characaterisation of the roots of Ehrhart

polynomials in the 3-dimensional case and we classify for

$n \leq 4$ all lattice polytopes whose roots of their

Ehrhart polynomials have all real part -1/2. These polytopes

belong to the class of reflexive polytopes.

**Keywords:**

Lattice polytopes, Ehrhart polynomial, reflexive polytopes