Hyperplane equipartitions plus constraints (Q1621410)

Over the years, equivariant topology methods, particularly, Borsuk-Ulam type theorems have been successfully used in topological combinatorics. However, the recently settled Topological Tverberg Conjecture suggests a need to go beyond Borsuk-Ulam type theorems. A famous remaining open question in topological combinatorics is the hyperplane mass equipartition problem, originating with Grünbaum in 1960 and generalized by Ramos in 1996: Question. What is the minimum dimension \(d:= \Delta(m;k)\) such that any \(m\) mass distributions \(\mu_1, \dots, \mu_m\) on \(\mathbb{R}^d\) can be simultaneously equipartitioned by \(k\) hyperplanes? Precise values of \(d\) 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 Blagojević, Frick, and Ziegler used for Tverberg-type results and recently to the above hyperplane mass equipartition problem, the author shows how the imposition of further conditions on the hyperplane arrangements themselves and/or the equipartition of additional masses by successively fewer hyperplanes yields a variety of optimal results for constrained equipartitions of \(m\) mass distributions in dimension \(U(m;k)\), including in dimensions below \(\Delta(m+1;k)\), which are still extractable via equivariance.