Hyperplane Equipartitions Plus Constraints

TL;DR

This paper develops a constraint method to improve hyperplane equipartition results in geometric combinatorics, achieving optimal bounds and exact values for constrained cases, including orthogonality and cascades, beyond traditional bounds.

Contribution

It introduces a novel constraint method that extends equivariant techniques to obtain optimal and exact hyperplane equipartition results under various constraints.

Findings

Achieves exact values for orthogonal hyperplane equipartitions.

Provides bounds below the classical dimension $oldsymbol{ extit{ ext{Delta}}}(m+1;k)$.

Extends equipartition results to constrained scenarios like cascades and prescribed conditions.

Abstract

While equivariant methods have seen many fruitful applications in geometric combinatorics, their inability to answer the now settled Topological Tverberg Conjecture has made apparent the need to move beyond the use of Borsuk--Ulam type theorems alone. This impression holds as well for one of the most famous problems in the field, dating back to 1960, which seeks the minimum dimension $d:=\Delta(m;k)$ such that any $m$ mass distributions in $\mathbb{R}^d$ can be simultaneously equipartitioned by $k$ hyperplanes. Precise values of $\Delta(m;k)$ have been obtained in few cases, and the best-known general upper bound $U(m;k)$ typically far exceeds the conjectured-tight lower bound arising from degrees of freedom. Following the "constraint method" of Blagojevi\'c, Frick, and Ziegler originally used for Tverberg-type results and recently to the present problem, we show how the imposition of further conditions -- on the hyperplane arrangements themselves (e.g., orthogonality, prescribed flat containment) and/or the equipartition of additional masses by successively fewer hyperplanes ("cascades") -- yields a variety of optimal results for constrained equipartitions of $m$ mass distributions in dimension $U(m;k)$, including in dimensions \textit{below} $\Delta(m+1;k)$, which are still extractable via equivariance. Among these are families of exact values for full orthogonality as well as cascades which maximize the "fullness" of the equipartition at each stage, including some strengthened equipartitions in dimension $\Delta(m;k)$ itself.