A Method of Deducing $L$-Polyhedra for $n$-Lattices

We suggest a method for selecting an $L$-simplex in an $L$-polyhedron of an $n$-lattice in Euclidean space. By taking into account the specific form of the condition that a simplex in the lattice is an $L$-simplex and by considering a simplex selected from an $L$-polyhedron, we present a new method for describing all types of $L$-polyhedra in lattices of given dimension $n$. We apply the method to deduce all types of $L$-polyhedra in $n$-dimensional lattices for $n=2,3,4$, which are already known from previous results.