On Polar Cones and Differentiability in Reflexive Banach Spaces

Let $X$ be a  Banach  space, $Csubset X$  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 ${mathop{rm int}}(mathrm{dom} sigma_C) neqemptyset$.  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 ${mathop{rm int}}(mathrm{dom}  sigma_C)$.