Orthogonal Ham-Sandwich Theorem in ℝ³

Abstract

The ham-sandwich theorem states that, given ≥ 2 measures in ℝ , it is possible to divide all of them in half with a single ( − 1)-dimensional hyperplane. We study an orthogonal version of the ham-sandwich theorem and define an orthogonal cut using at most 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 3 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 3 can be simultaneously bisected by an orthogonal cut. We present an algorithm for computing an orthogonal ham-sandwich cut in ( log ) time.