  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.
