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A Takahashi convexity structure on the Isbell-convex hull of an asymmetrically normed real vector space

Philani Rodney Majozi, Mcedisi Sphiwe Zweni

Abstract

Let $(X,\|\cdot\|)$ be an asymmetrically normed real vector space and let $\mathcal{E}(X,\|\cdot\|)$ denote its Isbell-convex (injective) hull viewed as a space of minimal ample function pairs. We introduce a canonical $T_{0}$-quasi-metric $q_{\mathcal{E}}$ on $\mathcal{E}(X,\|\cdot\|)$ of sup-difference type and show that the canonical embedding $i:X\to\mathcal{E}(X,\|\cdot\|)$ is isometric. Using the vector space operations on the hull, we define a barycentric map \[ \mathbb{W}(f,g,λ)=λf\oplus(1-λ)g,\qquad f,g\in\mathcal{E}(X,\|\cdot\|),\ λ\in[0,1], \] and prove that $(\mathcal{E}(X,\|\cdot\|),q_{\mathcal{E}},\mathbb{W})$ is a convex $T_{0}$-quasi-metric space in the sense of Künzi and Yildiz. For the standard affine convexity on $X$ we establish the equivariance $i(W(x,y,λ))=\mathbb{W}(i(x),i(y),λ)$, hence $i(X)$ is $\mathbb{W}$-convex in the hull. We further record stability properties of $W$-convex function pairs under hull operations and develop a Chebyshev-center/normal-structure framework on $\mathcal{E}(X,\|\cdot\|)$ yielding fixed point theorems for nonexpansive self-maps on bounded, doubly closed, $\mathbb{W}$-convex subsets of the hull.

A Takahashi convexity structure on the Isbell-convex hull of an asymmetrically normed real vector space

Abstract

Let be an asymmetrically normed real vector space and let denote its Isbell-convex (injective) hull viewed as a space of minimal ample function pairs. We introduce a canonical -quasi-metric on of sup-difference type and show that the canonical embedding is isometric. Using the vector space operations on the hull, we define a barycentric map \[ \mathbb{W}(f,g,λ)=λf\oplus(1-λ)g,\qquad f,g\in\mathcal{E}(X,\|\cdot\|),\ λ\in[0,1], \] and prove that is a convex -quasi-metric space in the sense of Künzi and Yildiz. For the standard affine convexity on we establish the equivariance , hence is -convex in the hull. We further record stability properties of -convex function pairs under hull operations and develop a Chebyshev-center/normal-structure framework on yielding fixed point theorems for nonexpansive self-maps on bounded, doubly closed, -convex subsets of the hull.
Paper Structure (25 sections, 15 theorems, 98 equations)

This paper contains 25 sections, 15 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quasi-metrics and convexity structures
  4. Asymmetric norms and the induced quasi-metric
  5. The Isbell-convex hull
  6. Algebraic operations on $\mathcal{E}(X,\|\cdot\|)$
  7. A canonical quasi-metric on $\mathcal{E}(X,\|\cdot\|)$ and a lifted convex structure
  8. A canonical $T_0$-quasi-metric $q_{\mathcal{E}}$ on $\mathcal{E}(X,\|\cdot\|)$
  9. Hull operations and compatibility with the embedding
  10. Definition of the lifted convex structure $\mathbb{W}$ on the hull
  11. Intertwining with the underlying convexity structure
  12. $W$-convex function pairs on $X$ and stability under hull operations
  13. Definition and first examples
  14. Minimal pairs are automatically $W$-convex
  15. Stability under hull operations
  16. ...and 10 more sections

Key Result

Proposition 2.3

If $(X,q,W)$ is a convex quasi-pseudometric space, then $(X,q^t,W)$ is convex. Moreover $(X,q^s,W)$ is a convex metric space in the sense of Takahashi.

Theorems & Definitions (67)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 57 more