  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.
