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

Abstract

Künzi 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 $\mathcal{E}(X,\Vert \cdot \Vert )$, we define a lifted convexity structure \[ \mathbb W(f,g,λ)=λf\oplus(1-λ)g \] and show that $(\mathcal{E}(X,\Vert \cdot \Vert ),q_{\mathcal{E}},\mathbb{W})$ 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.

29 pages