Geometric graphs and applications

几何图是指其顶点和边分别用点和(度量)线段标识的图。已知的例子是Delaunay三角剖分或多面体的1-骨架。作为抽象关联结构的“几何化”，这一概念在许多学科中有许多应用，如最优化(位置科学中的“树”)、离散几何(例如，度量极值点集上的ErdöS类型问题)、凸性(例如，多面体理论或恒宽体)，以及各种非欧几里得几何(例如，一般赋范空间)。通过这个“Neuantrag”，H.Martini教授(申请人，德国开姆尼茨工业大学)与A.Kamal教授(申请人，阿布迪斯耶路撒冷/巴勒斯坦阿尔克斯大学的大学教师)和Yaakov S.Kupitz教授(申请人，耶路撒冷/以色列希伯来大学讲师)一起申请一个涉及几何图形及其应用的三方研究项目。我们之所以选择这一主题，是因为我们都在这一领域工作了多年，并共同发表了相关论文。因此，我们已经开发了图论和几何方法的广泛混合，这是有希望的，以期真正达到所描述的研究计划的目标。我们已发表的结果主要涉及几何图的基本性质及其应用。现在，在我们的项目中，我们希望保持这样的研究方向，并希望在三年内就以下主题撰写八份联合出版物。首先，关于几何图的基本性质，我们将研究几类关于各种性质是极值的几何图，其中着色也将发挥作用。其次，在应用方面，我们将研究具有极直径图的有限点集(即典型的ErdöS问题)、最优化问题(位置科学)和有限点集的广义划分问题。所有这八篇论文都将是四位作者的论文(外部合著者为M.A.珀尔斯)。在应用中，我们还描述了第二个联合三年周期(通过Fortsetzungsantrag计划)，它将更具应用性质，主题包括：有限维实Banach空间中度量极值点集的构造以及几何图在位置科学(即优化)中的进一步应用。基于这一切，我们希望，随着对几何图的深入研究，我们的研究项目将在离散和计算几何、组合学和图论、凸性以及Minkowski几何等数学领域之间建立联系并创建新的或扩展现有的交互。此外，计划中的项目将对我们研究小组的工作产生积极影响(例如，在相互访问时在研究研讨会上讲课，以及关于三所大学博士生的裁判活动)。

Geometric graphs are graphs whose vertices and edges are identified with points and (metric) line segments, respectively. Known examples are Delaunay triangulations or 1-skeletons of polytopes. As "geometrization" of an abstract incidence structure, the notion has many applications, in disciplines like optimization ("trees" in location science), discrete geometry (e.g., Erdös-type problems on metrically extremal point sets), convexity (for instance, polytope theory or bodies of constant width), and also in various non-Euclidean geometries (e.g., in general normed spaces). With this "Neuantrag", Prof. H. Martini (Applicant, TU Chemnitz/Germany) applies together with Prof. A. Kamal (Applicant, University teacher at Alquds University, Abu Dis Jerusalem/Palestine), and Prof. Yaakov S. Kupitz (Applicant, Lecturer at the Hebrew University in Jerusalem/Israel) for a trilateral research project referring to geometric graphs and their applications. We have chosen this subject since all of us work in this field already for many years and published jointly related papers. So we developed already a broad mixture of graph-theoretic and geometric methods which is promising in view of reaching really the goals of the described research program. Our already published results mainly refer to basic properties of geometric graphs as well as to applications of them. Now, within our project, we want to stay in such research directions, and within three years we want to write eight joint publications on the following topics. First, regarding fundamental properties of geometric graphs, we will investigate several classes of geometric graphs, which are extremal regarding various properties, where also colorings will play a role. Second, in applied direction we will study finite point sets with extremal diameter graphs (i.e., typical Erdös-type problems), problems in optimization (location science) and generalized partition problems for finite point sets. All these eight papers will be four-authors papers (with M. A. Perles as external coauthor). In the application we also describe the second joint three-years period (planned via "Fortsetzungsantrag"), which will be more of applied nature, with topics like: constructions of metrically extremal point sets in finite-dimensional real Banach spaces and further applications of geometric graphs in location science (i.e., optimization). Based on all this we hope that, with a deeper study of geometric graphs, our research project will forge links and create new or expand existing interactions between the mathematical fields of discrete and computational geometry, combinatorics and graph theory, convexity, as well as Minkowski geometry. Furthermore, the planned project will positively influence the work of our research groups (for example, lectures in research seminars when visiting each other, and refereeing activities regarding PhD students, with respect to all three universities).