Transversal generalizations of hyperplane equipartitions

TL;DR

This paper extends classical hyperplane equipartition theorems to multiple hyperplanes for families of convex sets under generalized intersection conditions, advancing solutions to the Grunbaum–Hadwiger–Ramos problem.

Contribution

It introduces new topological and combinatorial methods to generalize existing hyperplane transversality results, including a novel proof of Dolnikov's theorem.

Findings

Generalizes Dolnikov's theorem to multiple hyperplanes

Provides solutions to the Grunbaum–Hadwiger–Ramos measure problem for two hyperplanes

Establishes topological Radon-type intersection theorems

Abstract

The classical Ham Sandwich theorem states that any $d$ point sets in $\mathbb{R}^d$ can be simultaneously bisected by a single affine hyperplane. A generalization of Dolnikov asserts that any $d$ families of pairwise intersecting compact, convex sets in $\mathbb{R}^d$ admit a common hyperplane transversal. We extend Dolnikov's theorem by showing that families of compact convex sets satisfying more general non-disjointness conditions admit common transversals by multiple hyperplanes. In particular, these generalize all known optimal results to the long-standing Gr\"unbaum--Hadwiger--Ramos measure equipartition problem in the case of two hyperplanes. Our proof proceeds by establishing topological Radon-type intersection theorems and then applying Gale duality in the linear setting. For a single hyperplane, this gives a new proof of Dolnikov's original result via Sarkaria's non-embedding criterion for simplicial complexes.