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
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.
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Felsner, S., Morawe, N. (2003). Infeasibility of Systems of Halfspaces. In: Aronov, B., Basu, S., Pach, J., Sharir, M. (eds) Discrete and Computational Geometry. Algorithms and Combinatorics, vol 25. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55566-4_18
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DOI: https://doi.org/10.1007/978-3-642-55566-4_18
Publisher Name: Springer, Berlin, Heidelberg
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