标题： 外部循环多面体和幅度多面体的凸性 显示英文标题

标题： Exterior Cyclic Polytopes and Convexity of Amplituhedra

摘要： 幅形体是格拉斯曼流形中的半代数集。 我们研究幅形体的凸性和对偶性。 我们引入了一个称为\textit{可扩展的凸性}的实半代数集在任何嵌入射影簇中的凸性概念。 我们证明了$k=m=2$幅形体在三维射影空间的直线格拉斯曼流形中是可扩展凸的。 在此过程中，我们引入了一个称为\emph{外部循环多面体}的新多面体，它推广了循环多面体。 它等于幅形体在普吕克嵌入中的凸包。 我们对外部循环多面体进行了组合分析，包括它的面及其对偶。 最后，我们引入了\textit{可扩展的双幅度多面体}，它与外部循环多面体的对偶密切相关。 我们证明了$k=m=2$的对偶幅形体仍然是一个幅形体，其中外部矩阵数据通过扭映射发生了变化。 显示英文摘要

摘要： The amplituhedron is a semialgebraic set in the Grassmannian. We study convexity and duality of amplituhedra. We introduce a notion of convexity, called \textit{extendable convexity}, for real semialgebraic sets in any embedded projective variety. We show that the $k=m=2$ amplituhedron is extendably convex in the Grassmannian of lines in projective three-space. In the process we introduce a new polytope called the \emph{exterior cyclic polytope}, generalizing the cyclic polytope. It is equal to the convex hull of the amplituhedron in the Pl\"ucker embedding. We undertake a combinatorial analysis of the exterior cyclic polytope, its facets, and its dual. Finally, we introduce the \textit{(extendable) dual amplituhedron}, which is closely related to the dual of the exterior cyclic polytope. We show that the dual amplituhedron for $k=m=2$ is again an amplituhedron, where the external matrix data is changed by the twist map.