Generalizations of the YaoYao partition theorem and the central transversal theorem
Abstract
We generalize the YaoYao partition theorem by showing that for any smooth measure in $R^d$ there exist equipartitions using $(t+1)2^{d1}$ 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 BorsukUlam type theorem for Stiefel manifolds that allows us to generalize the central transversal theorem and prove results bridging the YaoYao partition theorem and the central transversal theorem.
 Publication:

arXiv eprints
 Pub Date:
 July 2021
 arXiv:
 arXiv:2107.06233
 Bibcode:
 2021arXiv210706233M
 Keywords:

 Mathematics  Combinatorics
 EPrint:
 18 pages, 3 figures