On Polar Cones and Differentiability in Reflexive Banach Spaces

Let X be a Banach space, C X be a closed convex set included in a well-based cone K, and also let σ C be the support function which is de ned on C. In this note, we rst study the existence of a bounded base for the cone K, then using the obtained results, we nd some geometric conditions for the set C, so that int(dom σ C) ̸= ∅. The latter is a primary condition for subdifferen- tiability of the support function σ C. Eventually, we study Gateaux differentiability of support function σ C on two sets, the polar cone of K and int(dom σ C).