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

TL;DR

This paper introduces a convex Takahashi structure on the Isbell-convex hull of an asymmetrically normed space, establishing its properties as a convex quasi-metric space and deriving fixed point theorems.

Contribution

It defines a canonical quasi-metric and barycentric map on the Isbell-convex hull, proving convexity and fixed point results for nonexpansive maps.

Findings

The hull admits a canonical $T_0$-quasi-metric $q_{\ ext{\mathcal{E}}}$.

The embedding $i$ is isometric and preserves convexity.

Fixed point theorems are established for nonexpansive maps on convex subsets.

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 \[ \mathbb{W}(f,g,\lambda)=\lambda f\oplus(1-\lambda)g,\qquad f,g\in\mathcal{E}(X,\|\cdot\|),\ \lambda\in[0,1], \] and prove that $(\mathcal{E}(X,\|\cdot\|),q_{\mathcal{E}},\mathbb{W})$ is a convex $T_{0}$-quasi-metric space in the sense of K\"unzi and Yildiz. For the standard affine convexity on $X$ we establish the equivariance $i(W(x,y,\lambda))=\mathbb{W}(i(x),i(y),\lambda)$, 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.