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

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 $\mathcal{E}(X,\mathrm{\parallel }\cdot \mathrm{\parallel })$, we define a lifted convexity structure $$\mathbb{W}(f,g,\lambda )=\lambda f\oplus (1-\lambda )g$$ and show that $(\mathcal{E}(X,\mathrm{\parallel }\cdot \mathrm{\parallel }),q_{\mathcal{E}},\mathbb{W})$ 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.