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unimodular lattice
It is proved that for any unimodular lattice Λ with homogeneous minimum L>amp;gt;0 and any set of real numbers α1, α2,..., αn there exists a point (y1, y2,..., yn) of Λ such that
      
We prove that the action of the orthogonal group of an integral, unimodular lattice whose rank and signature satisfy a certain indefiniteness condition acts transitively on pairs of mutually orthogonal elements with the same isometric invariants.
      
An example of a 32-dimensional even unimodular lattice
      
An example is constructed of a 32-dimensional even unimodular lattice without roots.
      
An effective description of (n-1) -dimensional faces is given for a Voronoi parallelohedron formed in an even unimodular lattice of dimension 32 without roots.
      
Let Λ be a unimodular lattice in ?2, μ a homogeneous minimum of Λ; let P(a,b)??2 be a rectangle with vertices at the points (a,0), ...(0,b), P(a, b)+X its image under the translation by a vector X ∈ ?2.
      
The set of its roots can be identified with the 26-dimensional even unimodular lattice II25,1 = II1,1.
      
 

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