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).