The active bijection in graphs, hyperplane arrangements, and oriented matroids, 1: The fully optimal basis of a bounded region

We consider here the bounded case of the active mapping, where bounded corresponds to bipolar orientations in the case of graphs, and refers to bounded regions in the case of real hyperplane arrangements, or of oriented matroids. In terms of activities, this is the uniactive internal case. We introduce fully optimal bases, simply defined in terms of signs, strengthening optimal bases of linear programming. An optimal basis is associated with one flat with a maximality property, whereas a fully optimal basis is equivalent to a complete flag of flats, each with a maximality property. The main results of the paper are that a bounded region has a unique fully optimal basis, and that, up to negating all signs, fully optimal bases provide a bijection between bounded regions and uniactive internal bases. In the bounded case, up to negating all signs, the active mapping is a bijection.