An extension of a theorem by Yao & Yao

Abstract

In this paper we study $N_d(k)$ the smallest positive integer such that any nice measure $\mu$ in ${\mathbb{R}}^d$ can be partitioned in $N_d(k)$ parts of equal measure so that every hyperplane avoids at least $k$ of them. A theorem of Yao and Yao \cite{YY1985} states that $N_d(1)\le 2^d$. Among other results, we obtain the bounds $N_d(2)\le 3\cdot 2^{d-1}$ and $N_d(1)\ge C\cdot 2^{d/2}$ for some constant $C$. We then apply these results to a problem on the separation of points and hyperplanes.

References in corpus (2)

• Balanced Convex Partitions of Measures in ${\mathbb{R}}^d$
• On the Yao-Yao partition theorem

Cited by in corpus (1)

• Coloring decompositions of complete geometric graphs