On Covering Smooth Manifolds with a Q-arrangement of Simplicies: An inductive Characterization of Q-matrices

This paper is concerned with a covering problem of smooth manifolds of dimension n − 1 by stitching 2 n n-simplices formed with 2 n-lists of points along their common (n − 1)-facets. The n-simplices are in bijective correspondence with the vertices of an n-dimensional hypercube; they could be degenerate and are allowed to overlap. We leverage the underlying inductive nature of the problem to give a (non-constructive) topological characterization. We show that for low dimensions such characterization reduces to studying the local geometry around the specific points serving to form the simplices, solving thereby the problem for n ≤ 3. This covering problem provides a geometric equivalent reformulation of a relatively old, yet unsolved, problem that originated in the optimization community: under which conditions on the n × n matrix M , does the so called linear complementarity problem given by w − M z = q, w, z ≥ 0, and w.z = 0, have a solution (w, z) for all vectors q ∈ R n. If the latter property holds, the matrix M is said to be a Q-matrix