发表机构
HUN-REN Alfréd Rényi Institute of Mathematics; ELTE Eötvös Loránd University, Budapest（匈牙利科学院阿尔弗雷德·雷尼数学研究所; 布达佩斯罗兰大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在三维空间中构造新的直径图和鲁洛多面体，证明任意2-连通强对合自对偶图均可作为鲁洛多面体骨架，并证实及加强相关猜想，应用涉及常宽体、Steinitz定理和Borsuk猜想。
AI 中文摘要
本文构造了 $\mathbb{R}^3$ 中直径图和鲁洛多面体的新例子，并对其组合结构给出了完整刻画。对于有限点集 $X\subset\mathbb{R}^d$，其直径图是以 $X$ 为顶点集的图，其中构成直径对的点对由一条边连接。Grünbaum、Heppes 和 Straszewicz 独立证明了 $X\subset \mathbb{R}^3$ 的直径图至多有 $2|X|-2$ 条边，回答了 Vázsonyi 的一个问题。他们的证明依赖于球多胞体。球多胞体 $\mathcal{B}(X)$ 是以 $X$ 中点为球心的单位球的交集。若这些中心构成具有 $2|X|-2$ 个直径对的族，我们称该球多胞体为鲁洛多面体。Kupitz、Martini 和 Perles 证明了鲁洛多面体的骨架必须是 2-连通的强对合自对偶图。他们猜想在简单 3-连通情形下这也是充分条件。我们不仅证实了这一猜想，还证明了任何 2-连通（不一定是简单的）强对合自对偶图都可以作为某个鲁洛多面体的骨架出现。为构造新的鲁洛多面体，我们构造了新的直径图。已知任何三维直径图都是射影平面的非二分四边形剖分的子图。我们证明其逆命题也成立，即对任何这样的图，我们都构造出一个直径实现。这也证实并加强了 Montejano、Pauli、Raggi、Roldán-Pensado 关于强对合自对偶图度量嵌入的猜想。该构造依赖于刚性理论的思想。我们还讨论了这些结果的一系列应用，例如构造常宽体以及与 Steinitz 定理和 Borsuk 猜想的联系。
英文摘要
In this paper we construct new examples of diameter graphs and Reuleaux polyhedra in $\mathbb{R}^3$, obtaining a full characterization of their combinatorial structure. For a finite set of points $X\subset\mathbb{R}^d$, its diameter graph is the graph on vertex set $X$ where pairs forming a diameter pair are connected by an edge. Grünbaum, Heppes and Straszewicz independently proved that the diameter graph of $X\subset \mathbb{R}^3$ has at most $2|X|-2$ edges, answering a question of Vázsonyi. Their proof relied on ball polytopes. The ball polytope $\mathcal{B}(X)$ is the intersection of the unit balls centered at the points of $X$. We call a ball polytope a Reuleaux polyhedron if the centers form a family with $2|X|-2$ diameter pairs. Kupitz, Martini and Perles showed that the skeleton of a Reuleaux polyhedron must be a 2-connected strongly involutive self-dual graph. They conjectured that in the simple 3-connected case this is also sufficient. We not only confirm this conjecture, but we show that any 2-connected (not necessarily simple) strongly involutive self-dual graph arises as the skeleton of a Reuleaux polyhedron. To construct the new Reuleaux polyhedra we construct new diameter graphs. It was known that any 3-dimensional diameter graph is a subgraph of a non-bipartite quadrangulation of the projective plane. We show that the reverse holds. That is, for any such graph we construct a diameter realization. This also confirms and strengthens a conjecture of Montejano, Pauli, Raggi, Roldán-Pensado on metric embeddings of strongly involutive self-dual graphs. The construction relies on ideas from rigidity theory. We also discuss a number of applications of these results, such as the construction of bodies of constant width and connections to Steinitz's theorem and Borsuk's conjecture.
Comments35 pages, 24 figures
