Generalizations of the Yao-Yao partition theorem and the central transversal theorem

Abstract

We generalize the Yao-Yao partition theorem by showing that for any smooth measure in R d there exist equipartitions using (t + 1)2 d−1 convex regions such that every hyperplane misses the interior of at least t regions. In addition, we present tight bounds on the smallest number of hyperplanes whose union contains the boundary of an equipartition of a measure into n regions. We also present a simple proof of a Borsuk-Ulam type theorem for Stiefel manifolds that allows us to generalize the central transversal theorem and prove results bridging the Yao-Yao partition theorem and the central transversal theorem.

Transversal generalizations of hyperplane equipartitions

“…Using the D 4 -equivariant transverse homotopy H : (S n ) * 2 ×I → U ⊕m 2 ⊕W 2 ⊕V 2 of Proposition 6.7, we may now prove Theorem 6.1 via a standard zero counting argument as in [24]. See also [16,20].…”