多面体赋范空间：结构定理与局部有限铺砌
Polyhedral normed spaces: the structural theorem and locally finite tilings
- Università Cattolica del Sacro Cuore（天主教撒克心脏大学）
- University of Granada（格拉纳达大学）
- Universität Innsbruck（因斯布鲁克大学）
- Politecnico di Milano（米兰理工大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明具有($\Delta$)性质的多面体赋范空间的单位球面被代数真面覆盖，并推广到(VI)-多面体情形，进而刻画了局部有限铺砌的存在性，推广了Fonf的结果。
AI中文摘要：
Fonf（1981年）的一个基本结果断言，每个多面体Banach空间的单位球面都被单位球的真面所覆盖。本文的主要目的是研究同一结果在多面体赋范空间中的有效性，并给出一些应用。我们的第一个主要结果是，如果$X$是一个具有性质($\Delta$)的多面体赋范空间，则其单位球面被代数真面所覆盖。由此可知，如果该空间还额外是(VI)-多面体的，则其单位球面被真正的真面所覆盖。我们还给出了几个反例，表明这些结果在强意义下是最优的。例如，我们证明了存在(V)-多面体赋范空间，其单位球完全没有代数真面，并且存在具有($\Delta$)的多面体赋范空间，其单位球面不被真面覆盖。此外，我们给出了一个多面体赋范空间的例子，使得单位球的强暴露点集在球面上稠密，以及一些关于多面体赋范空间边界的结果和反例。我们这些材料的主要应用涉及赋范空间的局部有限铺砌，我们证明了赋范空间允许局部有限铺砌（由有界凸体构成）当且仅当它允许一个具有($\Delta$)的(VI)-多面体范数。特别地，这样的铺砌存在于每个具有($\Delta$)的多面体Banach空间中，这推广了Fonf的一个结果。
英文摘要：
A fundamental result due to Fonf (1981) asserts that the unit sphere of every polyhedral Banach space is covered by true faces of the unit ball. The principal aim of our paper is to study the validity of the same result for polyhedral normed spaces and present some applications. Our first main result is that if $X$ is a polyhedral normed space with property ($Δ$), then its unit sphere is covered by algebraic true faces. It then follows that if the space is additionally (VI)-polyhedral, then its unit sphere is covered by genuine true faces. We also present several counterexamples showing that these results are optimal in a strong sense. For instance, we show that there exist (V)-polyhedral normed spaces whose unit ball doesn't have any algebraic true face at all, and that there exist polyhedral normed spaces with ($Δ$) whose unit sphere is not covered by true faces. Further, we give an example of a polyhedral normed space such that the set of strongly exposed points of the unit ball is dense in the sphere, and some results and counterexamples concerning boundaries of polyhedral normed spaces. Our principal application of this material involves locally finite tilings of normed spaces, and we show that a normed space admits a locally finite tiling (by bounded convex bodies) if and only if it admits a (VI)-polyhedral norm with ($Δ$). In particular, such a tiling exists in every polyhedral Banach space with ($Δ$), which generalises a result of Fonf.
