Block partitions: an extended view

Given a sequence $S=(s_1,\dots ,s_m)\in [0,1{]}^m$, a block $B$ of $S$ is a subsequence $B=(s_i,s_{i+1},\dots ,s_j)$. The size $b$ of a block $B$ is the sum of its elements. It is proved in [1] that for each positive integer $n$, there is a partition of $S$ into $n$ blocks $B_1,\dots ,B_n$ with $|b_i - b_j| \le 1$forevery$i, j$. In this paper, we consider a generalization of the problem in higher dimensions.