Abstract
Linear inequalities attracted the attention of mathematicians as early as the first half of the last century (Fourier [i], Ostrogradskii [2]) in connection with applications, in particular, applications in analytical mechanics. However, the development of the theory of linear inequalities began only at the end of the century (Minkowski [3]). In 1901Farkas [4] made a thorough investigation of systems of linear inequalities and then in 1908 Voronoi [5] investigated quadratic forms in integer variables. In this latter paper Voronoi posed the general problem of studying geometric properties of a convex polyhedral set. Systems of linear inequalities were later studied by Stiemke [6], Dines [7], Weyl [8], and other mathematicians. However, the intensity of studying linear inequalities remained low for a long time, and only at the beginning of the 1950s, when a wide range of possibilities for applying the theory of linear inequalities to economics had already been determined (see Kantorovich [9], Koopmans [i0]), did it rise significantly (in the main, apparently, due to the investigations of applied nature).
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