Document Type : Research Paper

Authors

1 Department of Mathematics, Faculty of Science, Sahand University of Technology, Tabriz, Iran.

2 Department of Mathematics, Faculty of Science, Sahand University of Technology, Tabriz, Iran.

Abstract

Let $X$ be a  Banach  space, $C\subset X$  be  a  closed  convex  set  included  in  a well-based cone $K$, and also let $\sigma_C$ be the support function which is defined on $C$. In this note, we first study the existence of a  bounded base for the cone $K$, then using the obtained results, we find some geometric conditions for the set  $C$,  so that ${\mathop{\rm int}}(\mathrm{dom} \sigma_C) \neq\emptyset$.  The latter is a primary condition for subdifferentiability of the support function $\sigma_C$. Eventually, we study Gateaux differentiability of support  function $\sigma_C$ on two sets, the  polar cone of $K$ and ${\mathop{\rm int}}(\mathrm{dom} \sigma_C)$.

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Main Subjects

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