L-polytopes, Even Unimodular Lattices, and Perfect Lattices

It is shown here that every L-polytope of an even unimodular lattice does not generate the lattice. It is given here the corrected formulation of a previous result of the author [3] on relations between extreme L-polytopes and perfect lattices. We prove here the following special case. If the square radius of the circumscribing sphere of an extreme L-polytope P of a lattice L is less than the minimal norm m of L, then the m-extension of P generates a perfect lattice. 1 Introduction Recall some notions of integral lattices and L-polytopes. Details see in [1]. An L-polytope of a lattice L is the convex hull of all lattice points lying on an empty sphere. An empty sphere in a lattice L of dimension n is such a sphere that there is no lattice point inside the sphere, and the lattice points lying on the sphere affinely generate the n-dimensional space. We call the radius of the empty sphere a radius of the inscribed Lpolytope. Let P be an L-polytope of a lattice L. We take the center of..