The duality of two kinds of representations of convex sets is studied: the tangential representation of a convex body and the representations of its polar or negative polar by means of their weak* exposed points. The equivalence of the representations is proved and a condition for their validity is obtained for individual sets (the case of arbitrary locally convex space) and for classes of sets (the case of Gâteaux differentiability locally convex space). Properties of Gâteaux differentiability locally convex spaces are studied and some examples of such spaces are given.
相似文献This paper shows that the product of a Gâteaux differentiability space and a separable Banach space is again a Gâteaux differentiability space.