Exterior Cyclic Polytopes and Convexity of Amplituhedra
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866911073863467008 |
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| author | Mazzucchelli, Elia Pratt, Elizabeth |
| author_facet | Mazzucchelli, Elia Pratt, Elizabeth |
| contents | 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 Plücker 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_17620 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Exterior Cyclic Polytopes and Convexity of Amplituhedra Mazzucchelli, Elia Pratt, Elizabeth Combinatorics High Energy Physics - Theory Mathematical Physics 52B40 (primary), 52B12 (secondary) 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 Plücker 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. |
| title | Exterior Cyclic Polytopes and Convexity of Amplituhedra |
| topic | Combinatorics High Energy Physics - Theory Mathematical Physics 52B40 (primary), 52B12 (secondary) |
| url | https://arxiv.org/abs/2507.17620 |