  Building on a recent characterization of tope graphs of Complexes of Oriented
Matroids (COMs), we tackle and generalize several classical problems in
Oriented Matroids (OMs), Lopsided Sets (aka ample set systems), and partial
cubes via Metric Graph Theory. Our first main result is that every element of
an OM from a class introduced by Mandel is incident to a simplicial tope, i.e,
such OMs 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. Further, we show that the mutation graph of uniform OMs of
order at most 9 are connected, thus confirming a stronger conjecture of
Cordovil-Las Vergnas in this setting.
  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. Using
this, we confirm Las Vergnas' conjecture for antipodal partial cubes (a class
much lager than OMs) of small rank or isometric dimension.
