Lifted Takahashi Convexity on the Isbell-Convex Hull of an Asymmetrically Normed Real Vector Space
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866917525878472704 |
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| author | Majozi, Philani Rodney Zweni, Mcedisi Sphiwe |
| author_facet | Majozi, Philani Rodney Zweni, Mcedisi Sphiwe |
| contents | 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 $\mathcal{E}(X,\|\cdot\|)$, we define a lifted convexity structure \[ \mathbb W(f,g,λ)=λf\oplus(1-λ)g \] 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_12101 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Lifted Takahashi Convexity on the Isbell-Convex Hull of an Asymmetrically Normed Real Vector Space Majozi, Philani Rodney Zweni, Mcedisi Sphiwe General Topology Functional Analysis 54E35, 54E15, 46B20, 46A40, 54H25, 47H10 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 $\mathcal{E}(X,\|\cdot\|)$, we define a lifted convexity structure \[ \mathbb W(f,g,λ)=λf\oplus(1-λ)g \] 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. |
| title | Lifted Takahashi Convexity on the Isbell-Convex Hull of an Asymmetrically Normed Real Vector Space |
| topic | General Topology Functional Analysis 54E35, 54E15, 46B20, 46A40, 54H25, 47H10 |
| url | https://arxiv.org/abs/2603.12101 |