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The concept of a nonoblate cone in a Banach space is one of the most important ideas in the theory of ordered normed linear spaces. In connection with the introduction, the new class of SH-spaces by Smirnov (the H-spaces as Souslin spaces earlier), the problem of clarifying the role of the concept of nonoblateness of a cone in such spaces arises naturally. In the present paper, we will obtain a theorem about the nonoblateness of a generating cone in an SH-space and demonstrate a series of its applications to questions of differentiability with respect to a cone and of the continuity of a positive operator. This will allow us to obtain a theorem on the existence of a saddle point of the Lagrange function for linear optimization problems in SH-spaces. 

nonoblate cones, locally convex spaces, Kuhn-Tacker theorem, compact differentiability, closed graph theorem.