发表机构
Instituto de Investigaciones y Estudios Superiores Económicos y Sociales, Universidad Veracruzana（经济与社会高等研究与教育学院，韦拉克鲁斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在有限维巴拿赫空间等价范数射影化空间中构造有限多面体模型，通过αE-范数和E-扇给出参数化，并证明对数畸变度量与希尔伯特射影度量等距，提供有限维近似。
AI 中文摘要
我们研究了有限维巴拿赫空间X上等价范数的射影化空间N'(X)（赋予对数畸变度量）内部的有限多面体模型。给定一个有限对称方向集E，我们引入了αE-范数类，定义为对称多面体的闵可夫斯基泛函，这些多面体的顶点位于由E指定的射线上。我们确定了使该参数化非冗余的容许权重α，并证明它们给出了相应有限方向模型N(E)的一一参数化。然后，我们通过引入完备的、对称的、单纯形的E-扇（这些扇编码了相关多面体范数为线性的锥形区域）来细化这一描述。对于固定的扇F，我们定义了相应的几何和坐标扇模型N(F)和R(F)，并证明它们作为锥体自然同构。在除以正标量乘法后，坐标模型与相应的范数模型等距：N'(E)和N'(F)上的对数畸变度量恰好由容许权重空间上的希尔伯特射影度量表示。最后，我们证明了这些有限模型的完备性结果，并展示了它们如何为N'(X)的度量几何提供有限维多面体近似。
英文摘要
We study finite polyhedral models inside the projectivized space of equivalent norms $\mathcal{N}'(X)$ on a finite-dimensional Banach space $X$, endowed with the logarithmic distortion metric. Given a finite symmetric direction set $E$, we introduce the class of $αE$-norms, defined as Minkowski functionals of symmetric polytopes whose vertices lie on the rays prescribed by $E$. We identify the admissible weights $α$ for which this parametrization is non-redundant and prove that they give a one-to-one parametrization of the corresponding finite direction model $\mathcal{N}(E)$. We then refine this description by introducing complete, symmetric, simplicial $E$-fans, which encode the conical regions on which the associated polyhedral norms are linear. For a fixed fan $\mathcal{F}$, we define the corresponding geometric and coordinate fan models $\mathcal{N}(\mathcal{F})$ and $\mathcal{R}(\mathcal{F})$, and show that they are naturally isomorphic as cones. After quotienting by positive scalar multiplication, the coordinate models become isometric to the corresponding norm models: the logarithmic distortion metric on $\mathcal{N}'(E)$ and $\mathcal{N}'(\mathcal{F})$ is represented exactly by the Hilbert projective metric on the admissible weight spaces. Finally, we prove completeness results for these finite models and show how they provide finite-dimensional polyhedral approximations to the metric geometry of $\mathcal{N}'(X)$.
Comments28 pages. Submitted for publication
