Our main result implies the following easily formulated statement. The set of edges E of every finite bipartite graph can be split into poly(log |E|) subsets so the all the resulting bipartite graphcs are almost regular. The latter means that the ratio between the maximal and minimal non-zero degree of the left nodes is bounded by a constant and the same condition holds for right nodes. We prove a similar statement for n-dimensional sets and show how it can be used to relate inequalities for Shannon entropy of random variables to inequalities between sizes of sections and their projections of multi-dimensional finite sets.