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On dentability and cones with a large dual

Fernando García-Castaño, M. A. Melguizo Padial

Abstract

In this paper, we provide some equivalences on dentability in normed spaces. Among others we prove: the origin is a denting point of a pointed cone $C$ if and only if it is a point of continuity for such a cone and $\overline{C^*-C^*}=X^*$; $x$ is a denting point of a convex set $A$ if and only if $x$ is a point of continuity and a weakly strongly extreme point of $A$. We also analize how our results help us to shed some light on several open problems in the literature.

On dentability and cones with a large dual

Abstract

In this paper, we provide some equivalences on dentability in normed spaces. Among others we prove: the origin is a denting point of a pointed cone if and only if it is a point of continuity for such a cone and ; is a denting point of a convex set if and only if is a point of continuity and a weakly strongly extreme point of . We also analize how our results help us to shed some light on several open problems in the literature.
Paper Structure (4 sections, 17 theorems, 5 equations)

This paper contains 4 sections, 17 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Results on dentability
  3. Some open problems solved for cones in $\mathrm{q}\mathcal{G}^*$
  4. The geometry of some canonical order cones

Key Result

theorem 1

(From GARCIACASTANO20151178) Let $X$ be a normed space and $C \subset X$ a pointed cone. The following are equivalent: $0_X$ is a denting point of $C$. $C$ has a bounded slice. The interior of $C^*$ in $X^*$ is not empty.

Theorems & Definitions (31)

  • theorem 1
  • theorem 2
  • lemma 1
  • lemma 2
  • proof
  • lemma 3
  • proof
  • lemma 4
  • proof
  • proposition 1
  • ...and 21 more