Lifted Takahashi Convexity on the Isbell-Convex Hull of an Asymmetrically Normed Real Vector Space

Abstract

K\"unzi and Yildiz introduced convexity structures in the sense of Takahashi for $T_{0}$-quasi-metric spaces. In this article, we continue this line of study on the Isbell-convex hull of an asymmetrically normed real vector space. Using the canonical hull quasi-metric and the vector-space operations on $\mathcal{E}(X,\|\cdot\|)$, we define a lifted convexity structure \[ \mathbb W(f,g,\lambda)=\lambda f\oplus(1-\lambda)g \] and show that $(\mathcal{E}(X,\|\cdot\|),q_{\mathcal{E}},\mathbb W)$ is a convex $T_{0}$-quasi-metric space. We further prove compatibility with the canonical embedding and relate the construction to $W$-convexity of minimal function pairs.