How many atoms can be defined by boxes

, We study the function b(n, d), the maximal number of atoms defined by n d-dimensional boxes, i.e. parallelepipeds in the d-dimensional Euclidean space with sides parallel to the coordinate axes. We characterize extremal interval families defining b (n, 1) = 2n -1 atoms and we show that b(n, 2)=2n2 -6n+ 7. We prove that for every d, d b* (d)= lim b(n, d)jnd exists and 1 ~(d/2) Jl b* (d) "§.e. n-co Moreover, we obtain b*(3)=8/9.