Some Notes on the Differentiability of the Support Function
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Some Notes on the Differentiability of the Support Function Sima Hassankhali1 · Ildar Sadeqi1 Received: 26 October 2018 / Revised: 16 October 2019 / Accepted: 26 October 2019 © Iranian Mathematical Society 2020
Abstract Let X be a Banach space, C be a nonempty closed convex set and σC be the support function of the set C. In this work, we give some necessary and sufficient conditions for the set C to satisfy int(domσC ) = ∅, which enable us to study the Frechet and Gateaux differentiability of support function on the set C. Keywords Recession cone · Bounded base · Gateaux and Frechet differentiability · Support function Mathematics Subject Classification 47L07
1 Introduction The problem of differentiability and subdifferentiability of a convex continuous function has been studied extensively during the recent 50/years. These issues are important in the theory of optimization and the geometry of the Banach spaces. The norm function plays a paramount role among all convex continuous functions, since it encodes information related to the differentiability properties of them [10]. In the recent years, the focus has progressively moved to specific convex functions called support functions which deserve to be studied in more detail, since they could play a fundamental role in recent developments of optimization and variational analysis (see [2]). Let X be a Banach space, X ∗ be its dual and C be a nonempty subset of X . The support function of C is an extended real- valued function on X ∗ defined by ¯ σC : X ∗ → R,
σC (x ∗ ) := supx∈C x ∗ (x).
(1.1)
Communicated by Massoud Amini.
B
Ildar Sadeqi [email protected] Sima Hassankhali [email protected]
1
Department of Mathematics, Sahand University of Technology, Tabriz, Iran
123
Bulletin of the Iranian Mathematical Society
In case that X = Rn (the n-dimensional Euclidean space), and C is a subset of Rn+ (the positive cone of Rn ), the support function is very important in economics and it is strongly related to the cost function: ¯ g : Rn+ → R,
g(x) := inf a∈C x(a).
In fact, all properties of the support function σC can be translated to corresponding properties of the cost function. See Refs. [19,20] and references therein for more details. This paper is organized as follows: In Sect. 2, we state some preliminary definitions and theorems. In Sect. 3, we give necessary and sufficient conditions on a nonempty closed convex set C so that int(domσC ) = ∅. In Sect. 4, we extend the results related to differentiability of support functions to infinite dimensional cases, in general. Consequently, some conditions are given to characterize the Gateaux and Frechet differentiability, concerning the extremal points.
2 Preliminaries In this paper, let X be a reflexive Banach space, X ∗ be its dual and C be a nonempty subset of X . Denote by affC, lin0 C, clC, intC, rintC, bdC, rbdC, convC, extC, expC and str − expC, the affine hull of C, the linear space parallel to affC, the closure of C, the interior of C, the relative interior of C (that is the interior of C wit
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