On Covering Euclidean Space with Q-arrangements of Cones

TL;DR

This paper explores covering Euclidean space with cones related to Q-matrices, providing geometric and algebraic characterizations, especially for dimension three, and analyzing the structure of uncovered regions.

Contribution

It offers a new geometric reformulation of the Q-matrix problem and characterizes Q-matrices algebraically for low dimensions, advancing understanding of their structure.

Findings

Characterization of maximal uncovered regions in 3D

Algebraic semi-algebraic set for Q-matrices in 3x3 case

Explicit criteria for Q-matrices based on subdeterminants

Abstract

This paper is concerned with a covering problem of Euclidean space by a particular arrangement of cones that are not necessarily full and are allowed to overlap. The problem provides an equivalent geometric reformulation of the solvability of the linear complementarity problem defining the class of Q-matrices. Assuming feasibility, we rely on standard tools from convex geometry to study maximal connected uncovered regions, we term \emph{holes}. We then use our approach to fully characterize the problem for dimension $3$, regardless of degeneracy. We further provide, for $n \leq 3$, an algebraic characterization for the class of Q-matrices. That is, we show that, $M$ is a Q-matrix if and only if its entries belong to an explicit semi-algebraic set (in dimension $9$) where all the involved polynomials are subdeterminants of $M$. We showcase the usefulness of such a characterization by generating $3$-by-$3$ Q-matrices with specific interesting properties on the involved cones.