An extension of a theorem by Yao & Yao

In this paper we study $N_d(k)$ the smallest positive integer such that any nice measure $\mu$ in $\R^d$ 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.