Exterior Cyclic Polytopes and Convexity of Amplituhedra

TL;DR

This paper introduces the exterior cyclic polytope and extendable convexity, analyzing the convexity and duality properties of amplituhedra, especially for the case k=m=2, with new geometric and combinatorial insights.

Contribution

It defines extendable convexity for semialgebraic sets, introduces the exterior cyclic polytope, and explores the convexity and duality of amplituhedra in this new framework.

Findings

The 2=2 amplituhedron is extendably convex in the Grassmannian.

The exterior cyclic polytope generalizes cyclic polytopes and equals the convex hull of the amplituhedron.

The dual amplituhedron for k=m=2 is also an amplituhedron with modified data via the twist map.

Abstract

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.