论文标题
覆盖欧几里得空间，并用圆锥形的Q分组
On Covering Euclidean Space with Q-arrangements of Cones
论文作者
论文摘要
本文通过特定的锥体布置不一定饱满并且可以重叠的圆锥体布置涉及欧几里得空间的涵盖问题。该问题提供了定义Q-膜类的线性互补问题的可溶性的等效几何重新印象。假设可行性，我们依靠凸几何形状的标准工具来研究最大连接的发现区域，我们称为\ emph {holes}。然后，我们使用我们的方法充分表征尺寸$ 3 $的问题，无论堕落如何。我们进一步为$ n \ leq 3 $提供了q-静脉阶级的代数表征。也就是说，我们表明，$ m $是Q-matrix，并且仅当它的条目属于明确的半代数集（以$ 9 $为单位），其中所有涉及的多项式均为$ m $的子确定剂。我们通过生成$ 3 $ - by- $ 3 $ q-matrices，在涉及锥体上具有特定有趣的属性来展示这种特征的有用性。
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.
