Infeasibility of Systems of Halfspaces

Abstract

An oriented hyperplane is a hyperplane with designated good and bad sides.The infeasibility of a cell in an arrangement $\overrightarrow \mathcal{A}$ of oriented hyperplanes is the number of hyperplanes with this cell on the bad side. With MinInf($\overrightarrow \mathcal{A}$) we denote the minimum infeasibility of a cell in the arrangement. A subset of hyperplanes of $\overrightarrow \mathcal{A}$ is called an infeasible subsystem if every cell in the induced subarrangement has positive infeasibility. With MaxDis($\overrightarrow \mathcal{A}$) we denote the maximal number of disjoint infeasible subsystems of $\overrightarrow \mathcal{A}$. For every arrangement $\overrightarrow \mathcal{A}$ of oriented hyperplanes

$$MinInf(\overrightarrow \mathcal{A} {\text{)}} \geqslant {\text{MaxDis(}}\overrightarrow \mathcal{A} {\text{)}}{\text{.}}$$

In this paper we investigate bounds for the ratio of the LHS over the RHS in the above inequality. The main contribution is a detailed discussion of the problem in the case d = 2, i.e., for 2-dimensional arrangements. We prove that $MinInf(\overrightarrow \mathcal{A} {\text{)}} \leqslant 2\cdot {\text{MaxDis(}}\overrightarrow \mathcal{A} {\text{)}}$, in this case. An example shows that the factor 2 is best possible. If an arrangement $\overrightarrow \mathcal{A}$ of n lines contains a cell of infeasibility n, then the factor can be improved to 3/2, which is again best possible. We also consider the problem for arrangements of pseudolines in the Euclidean plane and show that the factor of 2 suffices in this more general situation.