Equipartitions with Wedges and Cones.

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, whether we can bisect more than $d$ masses, if we allow for bisections with more general objects such as cones, wedges or double wedges. We answer this question in the affirmative by showing that with all of these objects, we can simultaneously bisect $d+1$ masses. For double wedges, we prove a stronger statement, namely that $d$ families of $d+1$ masses each can each by simultaneously bisected by some double wedge such that all double wedges have one hyperplane in common. The second question is, how many masses we can simultaneously equipartition with a $k$-fan, that is, $k$ half-hyperplanes in $\mathbb{R}^d$, emanating from a common $(d-2)$-dimensional apex. This question was already studied in the plane, our contribution is to extend the planar results to higher dimensions. All of our results are proved using topological methods. We use some well-established techniques, but also some newer methods. In particular, we introduce a Borsuk-Ulam theorem for flag manifolds, which we believe to be of independent interest.