Mass partitions by parallel hyperplanes via Fadell-Husseini index

Abstract: In this paper, we study a problem of mass partitions by parallel hyperplanes. Takahashi and Sober\'on conjectured an extension of the classical ham sandwich theorem: any $d+k-1$ measures in $\mathbb{R}^d$ can be simultaneously equipartitioned by $k$ parallel hyperplanes. We construct a configuration space -- test map scheme and prove a new Borsuk-Ulam-type theorem to show that the conjecture is true in the case when the Stirling number of second kind $S(d+k-1, k)$ is odd. This recovers exactly the parity condition obtained by Hubard and Sober\'on via different methods, reinforcing the possibility that this condition is both necessary and sufficient. Our proof relies on a novel computation of the Fadell-Husseini index.