Depth in an arrangement of hyperplanes (Q1302033)

A collection of \(n\) hyperplanes in \(\mathbb R^d\) forms a hyperplane arrangement. The depth of a point \(\theta\in\mathbb R^d\) is the smallest number of hyperplanes crossed by any ray emanating from \(\theta.\) The authors prove that for \(d = 2\) there always exists a point \(\theta\) with depth at least \(\lceil n/3\rceil.\) This theorem allows to obtain a rather surprising result, which is a counterpart to the Birch's one about a configuration of points in the plane [\textit{B.~J.~Birch}, Proc. Camb. Philos. Soc. 55, 289-293 (1959; Zbl 0089.38502)]: Consider \(n = 3m\) lines in \(\mathbb R^2,\) all with distinct slopes. Then the \(n\) lines can be partitioned into \(m\) triplets (\(i, j, k)\) so that the \(m\) closed triangles \(\bigtriangleup(l_i, l_j, l_k)\) have a nonempty intersection. For \(d\geq 3\) the authors conjecture that the maximal depth is at least \(\lceil n/(d + 1)\rceil.\) For arrangements in general position, an upper bound \(\lfloor(n+d)/2\rfloor\) on the maximal depth is also established. Finally, algorithms to compute points with maximal depth are discussed.