Transversal Generalizations of Hyperplane Equipartitions

Abstract

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 Grünbaum–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.