Contact Vectors of Point Lattices

The contact vectors of a lattice $L$ are vectors $l$ which are minimal in the $l^2$-norm l in their parity class. It is shown that, in the space of all symmetric matrices, the set of all contact vectors of the lattice $L$ defines the subspace $M(L)$ containing the Gram matrix $A$ of the lattice $L$. The notion of extremal set of contact vectors is introduced as a set for which the space $M(L)$ is one-dimensional. In this case, the lattice $L$ is rigid. Each dual cell of the lattice $L$ is associated with a set of contact vectors contained in it. A dual cell is extremal if its set of contact vectors is extremal. As an illustration, we prove the rigidity of the root lattice $D_n$ for $n\ge 4$ and the lattice $E_6^*$ dual to the root lattice $E_6$.