In the paper The Amplituhedron , Nima Arkani-Hamed and Jaroslav Trnka introduced the geometric object amplituhedron. It is defined as follows (see also the lecture notes).
Let $Z$ be a $(k+m)\times n$ real matrix with maximal minors positive. Let $\tilde{Z}: Gr_{k,n}^{\geq 0} \to Gr_{k,k+m}$ be a map given by $A \mapsto AZ^t$. The tree amplituhedron $\mathcal{A}_{n,k,m}$ is the image $\tilde{Z}(Gr_{k,n}^{\geq 0}) \subset Gr_{k,k+m}$. Here $Gr_{k,n}^{\geq 0}$ is the totally non-negative Grassmannian: \begin{align} Gr_{k,n}^{\geq 0} = \{A \in Gr_{k,n}: \Delta_J(A) \geq 0, \forall J \in {[n] \choose k}\}, \end{align} where $\Delta_J(A)$ is the minor of $A$ using columns $J$.
What is the definition of loop amplituhedrons? Thank you very much.
