Corners and simpliciality in oriented matroids and partial cubes.

Building on a recent characterization of tope graphs of Complexes of Oriented Matroids (COMs), we tackle and generalize several classical problems in Oriented Matroids, Lopsided Sets (aka ample set systems), and partial cubes via Metric Graph Theory. These questions are related to the notion of simpliciality of topes in Oriented Matroids and the concept of corners in Lopsided Sets arising from computational learning theory. Our first main result is that every element of an Oriented Matroid from a class introduced by Mandel is incident to a simplicial tope, i.e, such Oriented Matroids contain no mutation-free elements. This allows us to refute a conjecture of Mandel from 1983, that would have implied the famous Las Vergnas simplex conjecture. The second main contribution is the introduction of corners of COMs as a natural generalization of corners in Lopsided Sets. Generalizing results of Bandelt and Chepoi, Tracy Hall, and Chalopin et al. we prove that realizable COMs, rank 2 COMs, as well as hypercellular graphs admit corner peelings. On the way we introduce the notion of cocircuit graphs for pure COMs and disprove a conjecture about realizability in COMs of Bandelt et al. Finally, we study extensions of Las Vergnas' simplex conjecture in low rank and order. We first consider antipodal partial cubes -- a vast generalization of oriented matroids also known as acycloids. We prove Las Vergnas' conjecture for acycloids of rank~3 and for acycloids of order at most 7. Moreover, we confirm a conjecture of Cordovil-Las Vergnas about the connectivity of the mutation graph of Uniform Oriented Matroids for ground sets of order at most 9. The latter two results are based on the exhaustive generation of acycloids and uniform oriented matroids of given order, respectively.