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Introductionbe a diophantine inequality defined for a given real > 1; hear , , , are real numbers with − ̸ = 0. H. Minkowski in his monograph [2] raise the question about minimum constant such that the inequality has integer solution other than origin. Minkowski with the help of his theorem on convex body has found a sufficient condition for the solvability of Diophantine inequalities in integers not both zero:But this result is not optimal, and Minkowski also raised the issue of not improving constant . For this purpose Minkowski has proposed to use the critical determinant. Given any set ℛ ⊂ R , a lattice Λ is admissible for ℛ (or is ℛ-admissible) if ℛ ⋂︀ Λ = ∅ or {0}. The infimum Δ(ℛ) of the determinants (the determinant of a lattice Λ is written (Λ)) of all lattices admissible for ℛ is called the critical determinant of ℛ. A lattice Λ is critical for ℛ if (Λ) = Δ(ℛ).Critical determinant is one of the main notion of the geometry of numbers [2,3,6].

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