University of MaraghehSahand Communications in Mathematical Analysis2322-580711120180801On Polar Cones and Differentiability in Reflexive Banach Spaces13233221510.22130/scma.2018.72221.284ENIldarSadeqiDepartment of Mathematics, Faculty of Science, Sahand University of Technology, Tabriz, Iran.0000-0001-5336-6186SimaHassankhaliDepartment of Mathematics, Faculty of Science, Sahand University of Technology, Tabriz, Iran.Journal Article20170926Let $X$ 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 ${\mathop{\rm int}}(\mathrm{dom} \sigma_C)$.https://scma.maragheh.ac.ir/article_32215_2e744dde303f4e6c175af724da107e48.pdf