Equipartitions with Wedges and Cones

Abstract

A famous result about mass partitions is the so called \emph{Ham-Sandwich theorem}. It states that any $d$ mass distributions in $\mathbb{R}^d$ can be simultaneously bisected by a single hyperplane. In this work, we study two related questions. The first one is how many masses we can simultaneously partition with a $k$-fan, that is, $k$ half-hyperplanes in $\mathbb{R}^d$, emanating from a common $(d-2)$-dimensional apex. This question was extensively studied in the plane, but in higher dimensions the only known results are for the case where $k$ is an odd prime. We extend these results to a larger family of values of $k$. We further present a new result for $k=2$, which generalizes to cones. The second question considers bisections with double wedges or, equivalently, Ham-Sandwich cuts after projective transformations. Here we prove that given $d$ families of $d+1$ point sets each, there is always a projective transformation such that after the transformation, each family has a Ham-Sandwich cut. We further prove a result on partitions with parallel hyperplanes after a projective transformation.