Orthogonal Ham-Sandwich Theorem in R

The ham-sandwich theorem states that, given d ≥ 2 measures in R, it is possible to divide all of them in half with a single (d − 1)-dimensional hyperplane. We study an orthogonal version of the ham-sandwich theorem and define an orthogonal cut using at most d hyperplanes orthogonal to coordinate axes. For example, a hyperplane orthogonal to a coordinate axis and the boundary of an orthant are orthogonal cuts. We prove that any three measures in R can be divided in half each with a single orthogonal cut. Applied to point measures, it implies that any three finite sets of points in R can be simultaneously bisected by an orthogonal cut. We present an algorithm for computing an orthogonal ham-sandwich cut in O(n log n) time.