قضیۀ بیشاب- فلپس در مخروط‌های نرم‌دار

در این مقاله مفهوم نقاط اتکاء مجموعه‌های محدب در مخروط‌های نرم‌دار معرفی شده و نشان داده می‌شود که در یک مخروط نرم‌دار پیوسته، تحت شرط‌های مناسب، مجموعه نقاط اتکاء مجموعه‌ای محدب اسکات بسته کران‌دار، ناتهی است. هم‌چنین قضیه بیشاب فلپس را برای مخروط‌های نرم‌دار بیان و اثبات می‌کنیم.

Bishop-Phelps Theorem for Normed Cones

IntroductionIn the last few years there is a growing interest in the theory of quasimetric spaces and other related structures such as quasinormed cones and asymmetric normed linear spaces, because such a theory provides an important tool in the study of several problems in theoretical computer science, approximation theory, applied physics, convex analysis and optimization. Many works on general topology and functional analysis have recently been obtained in order to extend the wellknown results of the classical theory of normed linear spaces to the framework of asymmetric normed linear spaces and quasinormed cones.‎An abstract cone is analogous to a real vector space‎, ‎except that we take  as the set‎ ‎of scalars‎. ‎ In 2004, O‎. ‎Valero introduced the normed cones and proved some closed graph and open mapping results for normed cones. Also Valero defined and studied some properties of quotient normed cones. P. Selinger studied the norm properties of a cone with its order properties and proved HahnBanach theorems in these cones under the appropriate conditions. Valero and his colleagues discussed the metrizability of the unit ball of the dual of a normed cone and the isometries of normed cones. Other properties are investigated in a series of papers by Romaguera, Sanchez Perez and Valero.  The BishopPhelps theorem is a fundamental theorem in functional analysis which has many applications in the geometry of Banach spaces and optimization theory. The classical BishopPhelps theorem states that “the set of support functionals for a closed bounded convex subset  of a real Banach space X, is norm dense in  and the set of support points of  is dense in the boundary of .  Indeed, E. Bishop and R. R. Phelps answer a question posed by ‎Victor Klee in 1958. We give an analogue to the normed cones‎, in fact we show that in a continuous normed cone the set of support points of a closed convex set is a dense subset of the boundary under the appropriate hypothesis.ConclusionIn this paper the notion of support points of convex sets in normed cones is introduced and it is shown that in a continuous normed cone, under the appropriate conditions, the set of support points of a bounded Scottclosed convex set is nonempty. We also present a BishopPhelps type Theorem for normed cones../files/site1/files/52/5.pdf