Symmetry, oriented matroids and two conjectures of Michel Las Vergnas

The paper has two parts. In the first part we survey the existing results on the cube conjecture of Las Vergnas. This conjecture claims that the orientation of the matroid of the cube is determined by the symmetries of the underlying matroid. The second part deals with Euclidean representations of matroids as geometric simplicial complexes defined by symmetry properties abstracting those of zonotopes. Both sections involve arguments concerning simplicial regions illustrating, once more, the fundamental importance of the simplex conjecture of Las Vergnas.