Depth in an arrangement of hyperplanes

Abstract

A collection of n hyperplanes in ℝd forms a hyperplane arrangement. The depth of a point θ ∈ ℝd is the smallest number of hyperplanes crossed by any ray emanating from θ. For d = 2 we prove that there always exists a point θ with depth at least ⌈n/3⌉. For higher dimensions we conjecture that the maximal depth is at least ⌈n/(d + 1)⌉. For arrangements in general position, an upper bound on the maximal depth is also established. Finally, we discuss algorithms to compute points with maximal depth.