带点质量的单不稳定凸多面体最小情形具有8个面和11个顶点
The smallest mono-unstable convex polyhedron with point masses has 8 faces and 11 vertices
AI总结：
本文利用半定优化生成不可行性证书，首次完整证明了带点质量的单不稳定凸多面体的最小情形为8个面、11个顶点，并推广到所有大于1的维数。
摘要：
在由John H. Conway于1966年发起的单静态多面体研究中，核心问题是构造具有最少面数和顶点数的此类物体。通过区分不同的材料分布和稳定性类型，这一问题扩展为一小族相关问题。尽管已经建立了关于所需面数和顶点数的许多上界和下界，但迄今为止这些问题均未被完全解决。本文改编了（Bozóki et al., 2022）中提出的算法，首次给出了该问题族中一个问题的完整答案：利用半定优化工具箱高效生成数十万个不可行性证书，我们首次证明了带点质量的单静态多面体的存在性，该多面体具有最少的顶点数（V=11）（定理3）和最少的面数（F=8）。我们还证明，在所有大于1的维数中，V=11是单不稳定多面体所能具有的最小顶点数（推论6）。
英文摘要：
In the study of monostatic polyhedra, initiated by John H. Conway in 1966, the main question is to construct such an object with the minimal number of faces and vertices. By distinguishing between various material distributions and stability types, this expands into a small family of related questions. While many upper and lower bounds on the necessary numbers of faces and vertices have been established, none of these questions has been so far resolved. Adapting an algorithm presented in (Bozóki et al., 2022), here we offer the first complete answer to a question from this family: by using the toolbox of semidefinite optimization to efficiently generate the hundreds of thousands of infeasibility certificates, we provide the first-ever proof for the existence of a monostatic polyhedron with point masses, having minimal number (V=11) of vertices (Theorem 3) and a minimal number (F=8) of faces. We also show that V=11 is the smallest number of vertices that a mono-unstable polyhedron can have in all dimensions greater than 1. (Corollary 6.)
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