On Polar Cones and Differentiability in Reflexive Banach Spaces
Journal: Sahand Communications in Mathematical Analysis (Vol.11, No. 1)Publication Date: 2018-08-01
Authors : Ildar Sadeqi; Sima Hassankhali;
Page : 13-23
Keywords : Recession cone; Polar cone; Bounded base; Support function; Gateaux differentiability;
Abstract
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 ${mathop{rm int}}(mathrm{dom} sigma_C)$.
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