Aenne Benjes: Type cones of Alcoved Polytopes

Abstract: For a fixed matrix $A$ with rows $a_1, \dots, a_n \in \mathbb{R}^d$ we consider the space of all right-hand sides $b \in \mathbb{R}^n$, such that $P_A(b) = \{x \in \R^d: Ax \le b\}$ is a non-empty polytope and all hyperplanes with normals $a_1, \dots,a_n$ are supporting. This space is known as `closed inner region’ or `closed irredundancy domain’ and carries a natural conic structure. It decomposes into so called `type cones’. Two right-hand sides $b,b' \in \R^n$ are contained in the same type cone, if the polytopes $P_A(b)$ and $P_A(b')$ are normally equivalent. In this talk, we want to have closer look on these structures in general, and assuming that the polytopes $P_A(b)$ are alcoved. This talk is based on joint work with Raman Sanyal and Benjamin Schröter.